Perturbation Methods, Bifurcation Theory and Computer Algebra (Applied Mathematical Sciences)

Richard H. Rand Dieter Armbruster

Perturbation Methods, Bifurcation Theory and Computer Algebra

With 10 Illustrations

Springer-Verlag New York Berlin Heidelberg London Paris Tokyo

Richard H. Rand Department of Theoretical & Applied Mechanics Cornell University Ithaca, New York 14853 USA

Dieter Armbruster Institut flir Informationsverarbeitung Universitat Tiibingen 74 Tiibingen I FRG

AMS Subject Classification: 34A34/68C20/34CXXl35B32/34DI 0170KXX170H05170H 15 Library of Congress Cataloging in Publication Data Rand, R. H. (Richard H.) Perturbation methods, bifurcation theory, and computer algebra. (Applied mathematical sciences; v. 65) Bibliography: p. Includes index. I. Perturbation (Mathematics) 2. Bifurcation theory. 3. Algebra-Data processing. 4. MACSYMA (Computer system) 1. Armbruster, Dieter. II. Title. III. Series: Applied mathematical sciences (SpringerVerlag New York Inc.) ; v. 65. 510 s 87-16703 QA1.A647 vol. 65 [QA871] [515.3'53] © 1987 by Springer-Verlag New York Inc. All rights reserved. This work may not be translated or copied in whole or in part without the wrillen permission of the publisher (Springer-Verlag 175 Fifth Avenue, New York, New York 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc. in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. Text prepared by the authors in camera-ready form. Printed and bound by R.R. Donnelley & Sons, Harrisonburg, Virginia. Printed in the United States of America.

9 87 6 5 4 3 2 I ISBN 0-387-96589-0 Springer-Verlag New York Berlin Heidelberg ISBN 3-540-96589-0 Springer-Verlag Berlin Heidelberg New York

Contents

Chapter I

Lindstedt's Method

Chapter 2

Center Manifolds

27

Chapter 3

Normal Forms

50

Chapter 4

Two Variable Expansion Method

89

Chapter 5

Averaging

107

Chapter 6

Lie Transforms

132

Chapter 7

Liapunov-Schmidt Reduction

155

Appendix

Introduction to MACSYMA

215

I

References

235

Index

239

Preface

Our purpose in writing this book is to provide computer algebra programs which implement a number of popular perturbation methods.

For each

perturbation method, we present an introduction to the method, a couple of example problems, sample runs of the computer algebra programs and complete program listings. In addition, we include examples of various elementary bifurcations, such as folds, pitchforks and Hopf bifurcations. problems.

These arise in the example

Specifically, we treat Hopf bifurcations in autonomous nonlinear

systems via Lindstedt's method, the construction of center manifolds for simple, degenerate and nilpotent bifurcations in ordinary differential equations, the determination of normal forms for Hopf bifurcations and Takens-Bogdanov bifurcations, and averaging for autonomous and nonautonomous systems.

Further, we use Lie transforms to determine normal forms in

Hamiltonian systems.

Bifurcation in partial differential equations, such as

reaction diffusion equations or the Bernard convection problem, are treated via Liapunov-Schmidt reduction. Moreover, we offer comparisons of the various methods.

We compare

averaging with normal forms, Liapunov-Schmidt reduction with center manifold reduction, Lindstedt's method with normal form calculations, and so on.

To

help in making the comparisons we frequently treat the same problem by two or more methods.

E.g., we derive the Hopf bifurcation formula both by Lindstedt's

method as well as via normal forms. Our motivation for applying computer algebra to perturbation problems comes from the nature of the computations involved

~n

these kinds of problems.

viii

The massive algebra usually required to obtain detailed results is more quickly and more accurately ~ccomplished by computer than by hand.

Since our emphasis

is on computation, we have dropped mathematical rigor in favor of intelligibility of the computational methods.

However, we have provided the

reader with references to standard mathematical textbooks or research papers. The book assumes a knowledge of mathematics through a first year graduate course in applied mathematics.

We have chosen the computer algebra

system MACSYMA because it is popular and easy to learn. and some familiarity with MACSYMA is desirable [35].

For the reader who has no experience with

MACSYMA we have provided a short introduction in the Appendix. This book is perhaps best read in front of a computer terminal running MACSYMA.

The reader could then enter the programs in this book as BATaI files.

and run them on the sample problems.

By examining the value of intermediate

variables, greater understanding can be gained as to how the methods and programs work.

Moreover. we hope that these programs will be useful utilities

to research workers in applied mathematics.

A note of caution has to be added:

As the computational compleXity of a problem is increased. e.g. by increasing the number of parameters or the number of equations or the order of truncation. there will come a point where the programs in this book will cease to work. either because of running out of memory or taking too long to run.

We suggest

that in such cases the reader may extend the usefulness of the program by tailorin~

it to fit the particular problem at hand.

We have tested the programs in this book on the following versions of MACSYMA: Eunice MACSYMA 308.2 on a VAX 8500 and MACSYMA 310.35 on a SYMBOLICS 3670.

The timings which are given at the end of each run are machine dependent

and approximate.

Even on the same machine. the time for a given run will vary

considerably due to "garbage collections" and other aspects of the LISP environment which are invisible to the user.

While we have tried to design the

programs to run efficiently. the inventive reader can probablY improve upon our schemes and is encouraged to do so.

ix

We offer to send an electronic file containing the programs to those readers who have access to BITNET.

Our BITNET addresses are currently:

RHRY€lCRNLVAX5 (for RHR) and URBY€lCRNLVAX5 (for DA). We wish to thank the following people for the help they gave us: Fritz Busse, Tapesh Chakraborty, P.Y. Chen. Vincent Coppola, Richard Cushman, Gerhard Dangelmayr. Larry Fresinsky. Ruediger Gebauer, Jim Geer, Mohammed Golnaraghi, John Guckenheimer, Werner Guettinger, Peter Haug, Tim Healey, Phil Holmes, Herbert Hui, Bill Keith, Jon Len, Alexander Mielke, Frank Moon, Martin Neveling, Ken Ridley, Jan Sanders. This work was partially supported by the Department of Theoretical and Applied Mechanics of Cornell University, the National Science Foundation, the Mathematical Sciences Institute at Cornell. the Deutsche Forschungsgemeinschaft and the Stiftung Volkswagenwerk.

Richard Rand DePartment of Theoretical and Applied Mechanics Kimball Hall Cornell University Ithaca, NY 14853 USA

Dieter Armbruster Institut fuer Informationsverarbeitung Universitaet Tuebingen Koestlinstrasse 6 7400 Tuebingen 1 West Germany May 1987 Ithaca

aIAPTER 1

LINDSTEDT'S METHOD

Introduction

Lindstedt's perturbation method is a classical scheme for obtaining approximate solutions to differential equations which contain a small parameter c.

The idea is to expand the solution in a power series in c,

(1)

x(t}

and to solve for the unknown functions xi(t} recursively, i.e., in the order xO(t}, x (t}, x 2 (t}, 1 Unfortunately, it is generally insufficient to simply substitute eq.(l} into the differential equation to be solved.

The shortcoming of such an

approach may be i 11ustrated by trying it on an example.

We wi 11 take van der

Pol's equation as our example:

(2)

For any c, this differential equation possesses a periodic solution called a limit cycle (unique to within an arbitrary phase shift in time ([8],§8.7}).

It's importance lies in the property that any initial condition

(besides the rest state x

= x' = O)

will eventually lead to the limit cycle,

LINDSTEDT'S METHOD

2

i.e .• the limit cycle is globally attracting (for c c

= O.

> 0).

Moreover, when

van der Pol's equation reduces to the simple harmonic oscillator and is

easy to solve.

Thus we may hope to "perturb off of" the e

=0

solution by

2 Keeping terms of O(e ). we

substituting the series (1) into equation (2). obtain:

Since we want a solution which is valid for all (small) values of e. we equate to zero each of the coefficients of en. for n n

= 0,1

= 0.1.2 •....

For

and 2 we obtain:

o

(4)

Since van der Pol's equation is autonomous (i.e .• bas no explicit time dependent terms), we can choose the instant t the limit cycle.

=0

Thus we can choose the initial condition x'(O)

loss of generality.

=0

without

From the expansion (1) we then obtain the initial

conditions:

(7)

to correspond to any point on

=0

LINDSTEDT'S METHOD

3

Eqs.(4) and (7) give

(8)

B cos t O

where B is as yet undetermined. O

Substituting (8) into (5) and using some trig

identities. we obtain

B 3

(9)

sin t +

~ sin 3t

which has the general solution:

[~

~

(10)

~.S1n 3t + Bl

t - BO] (- 2 cos t) - 32

cos t + Al sin t

where Al and B are arbitrary constants of integration. I Note the presence of the t cos t term in eq.(IO). unbounded as t

~

00.

This term grows

In the final expression for x. eq.(I). this term would

appear multiplied by c. i.e .• ct cos t.

Thus when t grows to be O(l/c). this

ct term becomes 0(1) and invalidates the assumption that each term of the series is asymptotically smaller than the preceding term.

Inclusion of such

unbounded secular or resonance terms prevents the obtained solution from being uniformly valid over the time interval [0. 00 ) .

More simplistically. since we

are looking for a periodic solution. we require all (nonperiodic) secular terms to be removed.

In the case of eq.(IO). this requires that we choose

2

(11)

So far so good.

The procedure fails. however. when we consider eq.(6).

First we apply the initial condition (7) to eq.(IO). giving:

LINDSTEDT'S METHOD

4

x (t) 1

(12)

=-

t

sin 3t +

~

sin t + B1 cos t

Substituting (8) and (12) into (6) and using some trig identities, we obtain

41

(13)

cos t + 2B

1

sin t -

23

cos 3t + 3B 1 sin 3t +

45

cos 5t

Note that the cos t term leads to secular behavior, and that no choice of the constant B will eliminate the problem. 1

Thus the foregoing method,

called regular perturbations, fails to yield a solution of the desired form. Lindstedt's method involves a small modification of regular perturbations, a modification which permits the removal of all secular terms. The idea of the modification can be explained physically by reference to a key distinction between linear and nonlinear dynamical systems.

Nonlinear behavior

is

(Think of the

chara~terized

by a dependence of frequency on amplitude.

plane pendulum, which obviously takes longer to complete a large amplitude oscillation than a small amplitude oscillation.)

In the case of linear

oscillations, however, the eigenfrequency is accompanied by an arbitrary amplitude, exemplified by the fact that eigenvectors are never unique but may be multiplied by an arbitrary constant. Based on this physical reasoning, Lindstedt's method permits the response of the nonlinear (6 # 0) system to occur at a different frequency from that of the linear (6

= 0)

system.

The idea is to stretch time with the

transformation: (14)

T

= w t,

where w, the frequency of the response, is expanded in a power series in 6:

(15)

LINDSTEDT'S METHOD where the constants k

i

are to be found.

5

From a strictly algebraic point of

view, (14) represents a change of independent variable which inserts the undetermined constants k

i

into the solution, and thereby permits the

troublesome secular terms to be removed. As an example, again consider van der Pol's eq.(2).

After the stretch

(14), (2) becomes:

2 2 d x

(16)

w

2

2

+ x + c (x

- 1)

W

dx dT

dT

=0

and after substituting (1) and (15) and collecting terms, we obtain the following equations (cf. eqs.(4)-(6»:

(18)

where primes now represent derivatives with respect to T. As in the case of regular perturbations, eq.(17) has the solution (8). For no secular terms in (18) we _find that B O given by (12).

(20)

=2

and k 1

= 0,

so that x 1 (T) is

When these results are substituted into (19), we obtain:

(4k + 2

+

4'5

1 4)

cos T + 2B

cos 5T

1

sin T -

23

cos 3T + 3B 1 sin 3T

LINDSTEDT'S METHOD

6

When eq.(20) is compared to its regular perturbation counterpart, eq.(13), we see tbat Lindstedt's method has permitted the removal of secular terms by the choice B1

=0

and k 2

= -1/16.

Computer Algebra

The application of Lindstedt's method to finding limit cycles in the class of equations

x"

(21)

+ x +

to

f(x,x')

=0

is conveniently accomplished by the use of computer algebra [35].

We present a

MACSYMA program which asks the user to enter the given function f(x,x') and the desired truncation order. listing.

Here is a sample run, followed by the program

LC is the name of the MACSYMA function which contains the program:

LC{); THE D.E. IS OF THE FORM: X" + X + E * F{X,X') ENTER F(X, Y), REPRESENTING X' AS Y Y*{X 2-1) ; 2 The d. e. is: x" + x + e ( (x - 1) y ) = 0 ENTER TRUNCATION ORDER 4; CHOICES FOR LIMIT CYCLE AMPLITUDE: 1) - 2 2) 2 A

3)

0

ENTER CHOICE NUMBER 2; Done with step 1 of Done with step 2 of Done with step 3 of Done with step 4 of

4 4 4 4

=0

LINDSfEJJf'S METHOD

x

=2

(sin(3 z) - 3 sin(z») e cos(z) - ----------------------4 2

(5 cos(5 z) - 18 cos(3 z) + 12 cos(z»

e

96

3

(28 sin(7 z) - 140 sin(5 z) + 189 sin(3 z) - 63 sin(z»

e

+ ---------------------------------------------------------- +

2304

4 17 e

2 e

w = ----3072

- --

[VAX 8500

TIME = 30 SEC.]

+ 1

16

Here is the MACSYMA program listing:

1* THIS PROGRAM APPLIES LINDSTEDT'S METHOD TO THE EQUATION:

x.. + X + E F(X,X') ASSUMING A LIMIT CYCLE EXISTS.

= 0,

CAlL IT WITH:

LCO;

LCO :=( 1* input the differential equation *1

kill(x,xlist,paramlist), print("THE D.E. IS OF THE FORM:

X"

+ X + E * F(X,X')

f:read("ENTER F(X,Y), REPRESENTING X' AS Y"), print("The d.e. is; x" + x + e (",f,") f;subst('DIFF(x,z,l)*w,y,f), 1* set up the series expansions *1

n:read("ENTER TRUNCATION ORDER"), w:1,

for i thru n do w:w+k[i]*eAi, x:b[O]*cos(z), xlist:[xx[O] = x],

= 0"),

= 0"),

LtNDSTEJJT'S METHOD

8

for i thru n do x:x+xx[i]*eAi,

1* plug into the d.e. and collect terms *1 depends (xx , z). templ:diff(x,z.2)+x/wA2+e*ev(f,diff)/wA2, templ:taylor(templ,e,O,n); for i thru n do eq[i]:coeff(templ.e,i),

1* set up pattern matching rules for use in solving d.e. *1 matchdeclare(nl,true), defrule(c.cos(nl*z).cos(nl*z)/(nl*nl-l», defrule(s.sin(nl*z),sin(nl*z)/(nl*nl-l». /* load poisson series package and set parameter */ outofpois(dummy). poislim: 100.

1* main loop */ for i:l thru n do block(

1* trigonometric simplificatibn *1 1* efficient alternative to EXPAND(TRIGREDUCE(EXPAND( ») */ templ:outofpois(ev(eqti].xlist,paramlist,diff», /* eliminate secular terms *1 if i

=1 then (paramlist:solve(coeff(templ,sin(z»,b[O]), print("CHOICES FOR LIMIT CYa..E AMPLITUDE:"). for j:l thru length(paramlist) do print(j.")

",part(paramlist,j,2»,

rl:read("ENTER CHOIcE NUMBER"). paramlist:appencl(solve(coeff(templ,cos(z»,k[l]), [part(paramlist.rl)]»

LINDSTEDT'S METHOD else

paramlist:append(paramlist, solve([coeff(templ.cos(z)),coeff(templ,sin(z))], [k[i],b[i-l]])),

templ:expand(ev(templ,paramlist)). xlist:expand(ev(xlist,paramlist)),

1* output progress *1 print("Done with step",i,"of",n),

1* exit here if last iteration *1 if i=n then go(end).

1* solve the d.e. *1 1* efficient alternative to ODE2() *1 templ:factor(ev(templ.xx[i] = 0)). templ:applybl(templ,c,s), templ:xx[i] = templ+a[i]*sin(z)+b[i]*cos(z),

1* fit the initial condition *1 temp2:rhs(templ). temp2:diff(temp2,z), temp2:solve(ev(temp2,z:0),a[i]). xlist:append(xlist. [ev(templ. temp2)]),

1* end of main loop *1 end),

1* output results *1 w:ev(w,paramlist), x:taylor(ev(x,xlist,paramlist).e,O,n-l), print("x =" ,x), print("w =" ,w))$

LINDSTEDT'S METHOD

10

In order to check this program, Figs.l and 2 compare the order 4 solution generated in the sample run with numerical results obtained by a Runge Kutta scheme. 6

In Fig.l there is reasonable agreement even though the value of

is as large as 1.

The solution has already become unacceptable by

6

= 2,

however (see Fig.2.)

Quadratic Nonlinearities

The foregoing program LC works well for functions f(x,x') (cf. eq.(21)) which, as in the case of van der Pol's equation, do not contain quadratic terms.

(22)

However, it fails on an example such as

x .. + x -

for the following reason.

o

x'(1 - x + x')

6

The program LC removes secular terms from the Xl

equation and asks the user for a choice of the limit cycle amplitude B on the O first pass through the perturbation method.

The problem here is that the

quadratic terms produce no secular terms on the first pass through the perturbation method.

This is due in essence to the identity

cos2t

= 2"1 +

1 cos 2t 2"

which contains no secular sin t or cos t terms. the program must wait until the

~

In order to handle such terms,

equation to find B ' i.e. until the second O

pass through the perturbation method. This modification can be facilitated by scaling the terms in the differential equation [36].

Let us take the d.e. in the general form:

LINDSTEDT'S METHOD

2.5

x(t)

o

t

8

-2.5

Fig.1. Numerical (=N) and Lindstedt 0(4) (=L) solutions for the limit cycle in van der Pol's eq.(2) for

= 1.

£

2.5

x(t)

o

t

8

-2.5

Fig.2. Numerical (= N) and Lindstedt 0(4) (= L) solutions for the limit cycle in van der Pol's eq.(2)for

£

=

2.

LINDSTEIJf'S METHOD

12

x .. + x + c x

+ b

+ d

1

1

x

3

+ other cubic terms

x 4 + other quartic terms

+ higher order terms

Now we scale x by setting u

=0

<=. X to obtain:

+ <=.2 (b

+ <=.3 (d

1

1

u

3

+ other cubic terms)

u4 + other quartic terms)

+ higher order terms

=0

There remains the question of how to scale the damping coefficient c. Let us expand c in a power series in <=.:

Now in order to perturb off of the simple harmonic oscillator we must take

Co = O.

Moreover, in order to remove resonant terms during the first pass, c 1

must be chosen to be zero.

(If c

1

were not chosen to be zero, the perturbation

method would fail to obtain a limit cycle.) term to be of order <=.2.

Thus we scale the linear damping

LINDSTEDT'S MUHOD

13

In summary, we require the terms to scale as follows: 2 linear: O(c ) quadratic: O(c) • cubic: O(c 2 ), 3

quartic: O(c ), etc.

For example in eqo(22). set c

c

1/2

~

and x

~

=c

~., + ~ - -;.2 ~'+ -;. ~ ~. _ -;. ~,2

(23)

x to obtain

=0

The following program LC2 is a generalization of LC for the system:

(24.1)

x'

-y + c f(x.y)

(24.2)

y'

x + c g(x,y)

where f(x,y) and g(x,y) are assumed to be in the form:

quadratic + c [linear + cubic] + c 2 quartic +

Here is a sample run based on eq.(23) in the form (24), followed by the program listing:

LC2() ; The doe. 's are of the form x' = -y + e*f, y' = x + e*g where f,g are of the form: quadratic + e*(linear + cubic) + e A 2*(quartic) + . Enter f: 0; f = 0

Enter g: E*Y-Y 2-X*Y; A

2 g

- y

- x y + e y

0

0

14

LINDSTEIJf'S METHOD

Enter truncation order: 4. Done with step 1 of 4 Choices for limit cycle amplitude: 1) 2) 3)

0 - 2 2

Enter choice number 3;

Done with step 2 of 4 Done with step 3 of 4 Done with step 4 of 4 x

=2

(4 sin(z) + 2 cos(2 z) - 2 sin(2 z) + 6) e cos(z) + -----------------------------------------3

+ (62 cos(z) - sin(z) + 32 cos(2 z) + 32 sin(2 z) + 3 cos(3 z) - 21 sin(3 z»

2 e /36 +

y

=2

(4 cos(z) - 4 cos(2 z) - 4 sin(2 z»

e

sin(z) 3

+ (cos(z) + 2 sin(z) - 64 cos(2 z) + 64 sin(2 z) + 63 cos(3 z) + 9 sin(3 z»

2 e /36 + .

4

w =

-

113 e ------

108

2

-

[VAX 8500 TIME

5 e ---- + 1

6

= 100 SEC.]

Here is the MACSYMA program listing:

LC2():=(

/* input the differential equation */ kill(x,y,xylist,paramlist), print("The d.e. 's are of the form x'

= -y

+ e*f. y

=x

+ e*g"),

print("where f,g are of the form: quadratic + e*(linear + cubic) + e"2*(quartic) + .,. "), f: read("Enter f: "), print("f =",f),

LINDSfEIJr'S MUROD

g:read("Enter g:"), print("g =" ,g),

1* set up the series expansions *1 n: read("Enter truncation order: "), k[O]: 1, k[l]:O, w:sum(k[i]*eAi,i,O,n), xy:[x:b[O]*cos(z)+sum(xx[i](z)*eAi,i,I,n), y:b[O]*sin(z)+sum(yy[i](z)*e Ai,i,l,n)], xylist:[xx[O](z)=b[O]*cos(z), yy[O](z)=b[O]*sin(z)],

1* plug into d.e.'s and collect terms *1 tempI: [-diff(x, z)*w-y+e*ev(f ,diff) ,-diff(y,z)*w+x+e*ev(g,diff)], temp2:taylor(templ,e,O,n), for i:l thru n do eq[i]:coeff(temp2,e,i),

1* main loop *1 for i:l thru n do block(

1* trigonometric simplification *1 temp3:expand(trigreduce(expand(ev(eq[i],xylist,paramlist,diff)»),

1* eliminate secular terms *1 if i=1 then (temp4:temp3,

go(skip~to-here-first_time»

else newparamlist: solve([coeff(part(temp3,1),sin(z»-coeff(part(temp3,2), cos(z», coeff(part(temp3,1) ,cos(z»+coeff(part(temp3,2) ,sin(z»], [b[i-2],k[i]]), if i=2 then (paramlist:newparamlist, print("Choices for limit cycle amplitude:"),

15

LINDSTEDT'S METHOD

16

for j:l thru length(paramlist) do print(j,")

",part(paramlist,j, 1.2)),

rl:read("Enter choice number"). paramlist:part(paramlist.rl)) else paramlist:append(paramlist.newparamlist). temp4:expand(ev(temp3.paramlist)). xylist:expand(ev(xylist.paramlist)). skip_to-here-first_time. /* output progress */

print("Done wi th step". i, "of", n), /* exit here if last iteration */

if i=n then go(end), /* solve the d.e. 's */

temp4a:subst(durnmy(z),yy[i](z).temp4), atvalue(durnmy(z).z=O.O). temp5:desolve(temp4a.[xx[i](z),dummy(z)]), tempSa:subst(yy[i](z).dummy(z),temp5), temp5b:subst(b[i].xx[i](O).tempSa), xylist:append(xylist,[temp5b]). /* end of main loop */

end) ,

/* output results */

w:ev(w,paramlist). soln:taylor(ev([x.y].xylist,paramlist),e,O,n-2). print("x =" .part(soln.l)). print("y =",part(soln,2)), print("w =",w))$

LINDSfEDT'S MErHOD

17

As a check on the program and the method, we compare the results of LC2 with numerical integration on eq.(22}.

Fig.3 shows the limit cycle of eq.(22)

for c = 0.16 (i.e. ~ = 0.4 in the perturbation scheme. see eq.(23.)}

The

curves in Fig.3 were obtained from the previous run (truncation of order 4). as well as from runs with truncation orders 6 and 8.

In addition. Fig.3 displays

numerical results generated by a Runge Kutta scheme. of

c,

Note how, for this choice

greater accuracy is achieved by taking more terms in the series.

Hopf Bifurcation

As an application of the program LC2. we investigate the Hopf bifurcation [36J.

Whenever an equilibrium point of the focus type (i.e. having

complex eigenvalues) changes its stability as a result of parameter changes in the differential equations. a limit cycle is generically born.

In order to use

Lindstedt's method to see how this occurs. we consider the most general version of eqs.(24) up to cubic terms. i.e. we expand f and g in Taylor series. also include damping terms with coefficient

~

so that varying

~

We

through zero

will change the stability of the equilibrium, producing the desired result:

(25.1)

x'

-y + c

(25.2)

y'

x + c

2

2

~

~

x + c

y + c

fxx

2

(~x

<no<

(~x

2

+ fxy x y +

+ gxy x y +

!xx.. 2 2 y)

~2

2

y)

LINDSTEIJf'S METHOD

18

12

x(t)

o

t 8

-12

Fig.3. Numerical (= N) and Lindstedt 0(8), 0(6) and 0(4) (=

L8, L6 and L4 respectively) solutions for the limit

cycle of eq.(22) for

E:

=

0.16 (Le.

e;=

0.4).

LINDSTEDT'S METHOD

19

Here is the run:

LC2() ;

The d.e. 's are of the form x' = -y + e*f, y' = x + e*g where f,g are of the form: quadratic + e*(linear + cubic) + e A2*(quartic) + ... Enter f: e*(fyyy*yA3/6+fxyy*x*yA2/2+fxxy*xA2*y/2+fxxx*xA3/6)+fyy*yA2/2+fxy*x*y +fxx*xA2/2+e*mu*x; 3 fyyy y

2 fxyy x y

2 fxxy x

3 y

fxxx x

e (------- + --------- + --------- + -------) + 6 2 2 6

f

2

2

fyy y

fxx x

------

+ fxy x y +

2

2 + e mu x

Enter g: e*(gyyy*yA3/6+gxyy*x*yA2/2+gxxy*xA2*y/2+gxxx*x~3/6)+gyy*yA2/2+gxy*x*y

+gxx*xA2/2+e*mu*y; 3

2 gxyy x Y

2

3

2

gxxy x

gxxx x gyy y Y e (------- + --------- + --------- + -------) + ------ + gxy x y + e mu y gyyy Y

g

6

2

2

6

2 2 gxx x + ------

2

Enter truncation order: 2;

Done with step 1 of 2 Choices for limit cycle amplitude: 1) 0 2) - 4 sqrt( - mu/(gyyy - gxy gyy + fyy gyy - gxx gxy + gxxy - fxx gxx + fxy f"Y'J + fxyy + fxx fxy + fxxx)) 3)

4 sqrt( - mu/(gyyy - gxy gyy + fyygyy - gxx gxy + gxxy - fxx gxx + fxy fyy + fxyy + fxx fxy + fxxx))

Enter choice number 3;

Done with step 2 of 2 x

=4

cos(z) sqrt(- mu/(gyyy + (- gxy + fyy) gyy - gxx gxy + gxxy - fxx gxx + fxy fyy + fxyy + fxx fxy + fxxx)) + . .

y

4 sin(z) sqrt(- mu/(gyyy + (- gxy + fyy) gyy - gxx gxy + gxxy - fxx gxx + fxy fyy + fxyy + fxx fxy + fxxx)) + . ..

LINDSTEDT'S METHOD

20 2

w= e

2

(2 gyy

2 + (5 gxx - 5 fxy) gyy - 3 gxyy + 2 gxy

2

- 3 gxxx + 5 gxx

+ (- fyy - 5 fxx) gxy

2 - fxy gxx + 3 fyyy + 5 fyy

2 + 5 fxx fyy + 2 fxy

+ 3 fxxy

2 + 2 fxx ) mu/(3 gyyy + (3 fyy - 3 gxy) gyy - 3 gxx gxy + 3 gxxy - 3 fxx gxx

+ 3 fxy fyy + 3 fxyy + 3 fxx fxy + 3 fxxx) + 1

= 331

[VAX 8500 TIME

SEC.]

The significance of this computation lies in the requirement that the limit cycle amplitude be real.

The limit cycle amplitude is given by the

expression:

limit cycle amplitude

(26)

=4

(_~S)I/2.

where S

(27)

= gyyy

+ gxxy + fxxx + fxyy + fyy gyy + fxy (fxx + fyy)

- fxx gxx -

gxy (gxx

+ gyy)

In particular this requires that the sign of S be opposite to the sign of~.

If S

=0

the computation is indecisive and offers no information.

See

Fig.4 for the two cases of subcritical and supercritical Hopf bifurcations, in which the stability of the limit cycle is respectively unstable and stable. Note that the form of the linear terms in eqs.(25) involves the symmetric appearance of

~

in both the x' and y' equations.

A general system of

the form

(28)

x'

=a

x + b Y + •••

c x + d y +

y'

can be transformed into

u'

=- n v

+

~

u +

v'

=nu

+

~

v + •••

LINDSTEDT'S METHOD

21

SUPERCRITICAL CASE, S < 0 y

y

x

x

j.l

< 0

j.l

> 0

A

STABLE LIMIT CYCLE STABLE

UNSTABLE EQUILIBRIUM

EQUILIBRIU~I

SUBCRITICAL CASE, S > 0 y

y

--+-f-f-HH-I---x

11 < 0

j.l

> 0

A

UNSTABLE LIMIT CYCLE STABLE EQUILIBRIUM

Fig.4. A

Supercritic~l

UNSTABLE EQUILIBRIUM

and subcritical Hopf Bifurcations.

= amplitude of limit cycle.

LINDSfEIJr'S MUROD

22

via the linear transformation [36]

(29)

x

where fl

= [- (d~a)2_

= b u,

[d;a] u - fl v

y:;:

bcf/2 and where /-L

= a;d

In this way we could apply the results (26),(27) to the system

(30)

by writing it in the form:

(31.1)

x'

=-

y

(31.2)

y'

=x

2 2 2 - c c y + c (a x - a x y + a y ) l 2 3

+ c

2

(b

l

x

3

- b

2

x

2

Y+ b

3

2 x Y - b

4

3 Y ) + •••

and then using the transformation (29) to get it in the form (25). Alternately we can apply the program LC2 directly to the system (31):

LC2() ;

The d.e.'s are of the form x' = -y + e*f, y' = x + e*g where f,g are of the form: quadratic + e*(linear + cubic) + e~2*(quartic) + ... Enter f: 0; f

=0 Enter g: -c*y*e+(al*x A2-a2*x*y+a3*y A2)+e*{bl*xA3-b2*xA 2*y+b3*x*yA2-b4*y A3); g

=e

3

(- b4 y

2

+ b3 x y

2

- b2 x

3

y + bl x ) + a3 y

2

2

- a2 x y - c e y + al x

LINDSTEDT'S METHOD

23

Enter truncation order: 2; Done with step 1 of 2 Choices for limit cycle amplitude: 1)

0

2 )

- 2 sqrt(- -------------------------) 3 b4 + b2 - a2 a3 - al a2

3)

2 sqrt(- -------------------------) 3 b4 + b2 .:. a2 a3 - al a2

c

c

Enter choice number 3; Done with step 2 of 2 x

=2

Y

c

cos(z) sqrt(- -------------------------) + . . 3 b4 + b2 - a2 a3 - al a2

c 2 sin(z) sqrt(- -------------------------) + . . 3 b4 + b2 - a2 a3 - al a2

2 (3 b3 + 9 bl - 4 a3

w

2 - 10 a1 a3 - a2

2 2 - 10 a1 ) c e

1 - ---------------------------------------------------18 b4 + 6 b2 - 6 a2 a3 - 6 a1 a2

[VAX 8500 TIME

74 SEC.]

From this run we can conclude that for small c eq.(30) possesses a limit cycle if the quantity

(32)

has the opposite sign to that of the damping c. van der Pol's equation (2), where c

= -I,

b

2

=1

For example, in the case of and the other coefficients are

zero, the quantity (32) becomes unity and the limit cycle amplitude is correctly evaluated as 2.

Note also that the c 2 contribution to the frequency

w vanishes for van der Pol's equation.

must be replaced by c

2

(Cf. eq.(15) where k

l

for comparison with the foregoing run.)

o and

where c

LINDSTEDT'S METHOD

24

Exercises

1.

Find an

equ~tion

of the form

x·' + x +

which

exhib~ts

=0

f(x,x')

Eo

a limit cycle with frequency w

=1

+ k

1

t + •••. where k

1

~

O.

Note that the examples used in this Chapter. namely van der Pol's equation (2).

= O.

eq.(22), and eq.(30), all have k 1 Hint:

2.

Take a

= a2 = a3 = 0

l

in eq.(30).

We have dealt exclusively with autonomous systems in this Chapter.

Write a

program to apply Lindstedt's method to nonautonomous equations of the form

x"

+ x +

f(x,x' ,t)

Eo

= O.

As an example, consider the forced damped Duffing equation,

x'· + x + t

where w = 1 + kIt + Discussion:

= wt

2 2 d x w -- + x +

Eo {j W -

and then we expand x T.

3

=t

'Y

cos wt

+ .••• is given.

First we set T

dT2

period 2v in

2

~t

x' + t a x

{j

to get

dx

+

Eo

a x

3

t

'Y

cos

T

dT

= XO(T)

+

X1 (T)

Eo

+ •••.

We look for solutions with

i.e. witp the same period as the forcing function.

25

LINDSfEIJf'S METHOD

In order to eliminate secular terms in the xl equation we can now choose A and B ' O O formulation.)

(Note that the ki's are viewed as being given in this

Similarly the complementary solution for the Xi equation can be

= Ai

chosen as Xi

sin

T +

comp

B cos i

T

and an appropriate selection of Ai and

B will eliminate secular terms in the x + equation. i l i Unfortunately the-algebraic equations on A and B (or more generally O O on Ai and B ) are usually too complicated to solve in closed form. i Transformation to polar coordinates

usually helps.

E.g. in the case of Duffing's equation. A and B satisfy O O

which become in polar coordinates,

which can be combined to give

,2

26

3.

LINDSfEIIf'S MEfHOD

Mathieu's equation.

x"

+ (6 +

to

cos t) X

is both linear and nonautonomous. 6

i

=0

The problem here is to find the coefficients

in the expansion

+ •••

n

= 0,1,2,···

such that the differential equation exhibits periodic solutions.

This

expansion represents a transition curve in the 6-6 parameter plane (hence the phrase parametric excitation) which separates regions of bounded behavior from regions of unbounded behavior.

See e.g. [44].

Write a program to accomplish this task.

Note that since the equation

is linear, no frequency-amplitude relation is expected and no stretch

T

=w t

is needed, so that regular perturbations will work on this problem. After expanding x then X

o

= cos

= Xo +

6

+ •••• separately choose X

~t. as each gives a distinct transition curve.

elimination of secular terms.

4.

Xl

o

= sin ~t

Find the 6

i

and by

See [35].

Apply the transformation (29) to eqs.(3l} to get them in the form (25).

Then show that the results (26) and (27) agree with results of the last run in the Chapter on eqs.(3l} directly.

CHAPTER 2

CENTER MANIFOLDS

Introduction

Center manifold theory [6] is a method which uses power series expansions in the neighborhood of an equilibrium point in order to reduce the dimension of a system of ordinary differential equations.

The method involves

restricting attention to an invariant subspace (the center manifold) which contains all of the essential behavior of the system in the neighborhood of the equilibrium point as t

~ 00.

This method is applicable to systems which, when linearized about an eqUilibrium point, have some eigenvalues which have zero real part. and others which have negative real part.

We assume that no eigenvalues have positive

real part. since in such a case the center manifold will not be attractive as t

~ 00.

Under such assumptions the components of the solution of the linearized equations which correspond to those eigenvalues with negative real part will decay as t

~oo.

and hence the motion of the linearized system will

asymptotically approach the space 8

1

spanned by the eigenvectors corresponding

to those eigenvalues with zero real part.

The center manifold theorem [6]

assures us that this picture (which is so far based on the linearized equations) extends to the full nonlinear equations, as follows:

CENTER MANIFOLDS

28

There exists a (generally curved) subspace S2 (the center manifold) which is tangent to the (flat) subspace Sl at the equilibrium point. and which is invariant under the flow generated by the nonlinear equations.

All

solutions which start sufficiently close to the equilibrium point will tend asymptotically to the center manifold.

In particular. the theorem states that

the stability of the equilibrium point in the full nonlinear equations is the same as its stability when restricted to the flow on the center manifold ([6J. p.4).

Moreover. any additional equilibrium points or limit cycles which occur

in a neighborhood of the given equilibrium point on the center manifold are guaranteed to exist in the full nonlinear equations ([6], p.29). Thus center manifold theory allows us to eliminate algebraic complications which are due to unessential behavior and to focus on the motion in the center manifold which contains the critical information concerning the system's local stability and bifurcation of steady state behavior.

In doing so

we omit reference to how the system gets onto the center manifold, but rather we are assured of the asymptotic stability of the center manifold.

Since the

eigenvalues associated with the linearized flow in the center manifold have zero real part, the study of the motion in the center manifold must still be accomplished and may not be an easy task.

In this Chapter we will be concerned

with the question of how to reduce a system to its center manifold and we will leave for later Chapters treatment of the flow on the center manifold. The method will be illustrated by reference to the following example [31J of a system of three differential equations:

(1)

x

(2)

y

(3)

z

=y

- x - x z

=-

z + a x

2

CENTER MANIFOLDS

29

Here we have a feedback control system in which the spring constant in the simple harmonic oscillator of eqs.(1),(2) is given by 1 + z.

The control

2 variable z is governed by eq.(3) and involves the feedback term a x .

We wish

to know the stability of the equilibrium at the origin. Although the system (1)-(3) is 3-dimensiona1, we can use center manifold theory to reduce the stability question to the study of a 2-dimensional system.

Linearizing about the origin, we see that the x-y plane

is associated with a pair of pure imaginary eigenvalues, while the z axis corresponds to an eigenvalue of -1.

Thus the center manifold is a

2-dimensional surface which is tangent to the x-y plane at the origin. In order to obtain an approximate expression for the center manifold, we write z as a function of x and y, and expand in a power series.

Since the

center manifold is tangent to the x-y plane, we begin the power series with quadratic terms:

(4)

z

a

2 ,O x

2

+ a 1 ,l x y + a O,2 y

2

+ •••

The requirement that the center manifold be invariant is satisfied by substituting (4) into (3):

(5)

(2 a 2 ,O x + a 1 ,l y) x

- (a2 ,O x

2

+ (2 a O,2 y + a 1 ,l x) y' + •••

2

+ a ,l x y + a ,2 y ) + a x

1

O

Note that (5) involves the derivatives x expressions for these from (1) and (2) gives:

2

+ •••

and y'.

Substitution of

30

(6)

CENTER

MANIFOLDS

(2 a 2 •0 x + a • y) y + (2 a O. 2 y + a l . 1 x) (- x - x z) + ••• = l 1 2

- (a2 •0 x

2 2 + a l • I x y + a O•2 y ) + a x + •••

This last step once again introduced z into the computation (since y' depended on z).

So we again eliminate z by substituting (4) into (6).

Collecting terms and neglecting cubic and higher order terms. this gives:

(7)

(~.O - a l . 1 -

a) x

2

+ (2 ~.O - 2 a O• 2 + a l • l ) x y

Equating the coefficients of xiyj equal to zero. and solving the three resulting equations for the a . . ·s. we obtain:

1.J

(8)

a 2 •0 =

3

5

a

a l •1

2

=- 5

a

a O. 2

= 25 a

Substitution in (4) gives the following first approximation ,to the center manifold:

(9)

z

= 53 a

x

2

2 2 2 - 5,a x y + 5 a y +

We may now substitute (9) into eqs.(1).(2) in order to obtain approximate equations for the flow on the center manifold:

(10)

x

= y

(11)

y'

=-

3 3 2 2 2 2 x - 5 a x + 5 a x y - 5 a x y + •••

CENTER MANIFOLDS

31

Although approximate. eqs.(10).(11) contain the answer to the question of the stability of the origin in the original system (1)-(3).

The point of

center manifold theory is that eqs.(10),(II) are easier to analyze than eqs.(I)-(3).

We will return to the analysis of eqs.(10).(II) in the next

Chapter on normal forms.

Computer Algebra

The calculation of center manifolds involves the manipulation of truncated power series and is readily performed using computer algebra. present a MACSYMA program to accomplish such computations. follows a sample run on the preceding example:

CM();

ENTER NO. OF EQS. 3;

ENTER DIMENSION OF CENTER MANIFOLD 2;

's MUST BE ARRANGED SO THAT TIlE FIRST 2 EQS. REPRESENT TIlE CENTER MANIFOLD. I.E. ALL ASSOCIATED EIGENVALUES ARE ZERO OR HAVE ZERO REAL PARTS.

TIlE D.E.

ENTER SYMBOL FOR VARIABLE NO. 1

X; ENTER SYMBOL FOR VARIABLE NO.2

Y; ENTER SYMBOL FOR VARIABLE NO. 3 Z;

ENTER ORDER OF TRUNCATION 2;

ENTER RHS OF EQ. 1 D x /DT

Y; ENTER RHS OF EQ. 2 DylDT= -X-X*Z; ENTER RHS OF EQ. 3 Dz/DT= -Z+ALPHA*X 2; A

We

The program listing

CENTER MANIFOLDS

32 dx

-- = Y dt dy

-- = -

x Z - x

dt dz

2

-- = alpha

x

- z

dt

CENTER MANIFOLD:

2 2 alpha

2 2 alpha x

y

= ---------- - -----------

[z

5

3 alpha x

y

+ ----------]

5

5

FLOW ON THE C.M. :

2 dx

[-- = y, dt

dy --

dt

=-

2 alpha y

2 2 alpha x y

3 alpha x x (---------- - ----------- + ----------) - x]

5

5

5

[VAX 8500 TIME = 6 SEC.] Here is the MACSYMA listing for the program CM:

CMO:={ Itt INPUT PROBLEM ttl

N:READ{"ENTER NO. OF EQS."). M:READ{"ENTER DIMENSION OF CENTER MANIFOLD"). PRINT{"THE D.E. 'S MUST BE ARRANGED SO THAT THE FIRST" .M, "EQS."), PRINT{"REPRESENT THE CENTER MANIFOLD.

I.E. ALL ASSOCIATED"),

PRINT{"EIGENVALUES ARE ZERO OR HAVE ZERO REAL PARTS. "), FOR I: 1 THRU N DO XCI] :READ{"ENTER SYMBOL FOR VARIABLE NO.", 1), L:READ("ENTER ORDER OF TRUNCATION"). FOR I: 1 THRU N DO ( PRINT("ENTER RHS OF EQ.". I), PRINT{"D" ,X[I],"/DT ="),

CENTER MANIFOLDS G[ I] :READ()), /*

SET UP D.E. 's */

FOR I: 1 THRU N DO DEPENDS(X[I].T). FOR 1:1 THRU N DO (EQ[I]:DIFF(X[I],T)=G[~].

PRINT(EQ[I])), /* FORM POWER SERIES */ SUB:MAKELIST{K[I].I,l.M), VAR:PRODUCT{X[I]AK[I].I.1.M), UNK: []. FOR P:M+1 THRU N DO{ TEMP:A[P,SUB]*VAR. FOR I: 1 THRU MDO TEMP:SUM{EV{TEMP.K[I]=J).J.O.L). TEMP2:TAYLOR(TEMP.MAKELIST{X[I].I.1.M).O.L). /* REMOVE CONSTANT AND LINEAR TERMS

*/

TEMP3:TEMP2-PART{TEMP2.1)-PART(TEMP2,2). SOLN[P]:EXPAND(TEMP3). /*

PREPARE LIST OF UNKNOWNS */

SETXT01:MAKELIST{X[I]=f.I.1.M). /*

TURN SUM INTO A LIST */

UNKN[P]: SUBST("[". "+" .EV(TEMP3.SETXT01». UNK:APPEND(UNK.UNKN[P]). SOL:MAKELIST(X[P]=SOLN[P].P.M+1.N). /*

SUBSTITUTE INTO D.E. 'S

*/

CMDE:MAKELIST(EQ[I].I,l.M). REST:MAKELIST(LHS{EQ[I])-RHS(EQ[I]),I.M+1,N). TEMP4:EV{REST.SOL.DIFF).

CENTER MANIFOLDS

34 TEMP5:EV(TEMP4,CMDE,DIFF). TEMP6:EV(TEMP5.SOL) •

TEMP7:TAYLOR(TEMP6.MAKELIST(X[I],I,I,M),O,L),

1* COLLECT TERMS *1 COUNTER: 1,

1* MAKE LIST OF TERMS *1 TERMS: SUBST(" [" • "+" ,SOLN[N]), TERMS:EV(TERMS,A[DUMMY,SUB]:=I). FOR I: 1 THRU N-M DO ( EXP[I]:EXPAND(PART(TEMP7.I». FOR J: 1 THRU LENGTII(TERMS) DO( COND[COUNTER] :RATCOEF(EXP[I], PART(TERMS , J». COUNTER: COUNTER+ 1) ) • CONDS:MAKELIST(COND[I], I ,I.COUNTER-I).

1* SOLVE FOR CENTER MANIFOLD *1 ACOEFFS: SOLVE(CONDS. UNK) • CENTERMANIFOLD: EV(SOL ,ACOEFFS) • PRINT("CENTER MANIFOLD:"). PRINT(CENTERMANIFOLD) •

1* GET FLOW ON eM *1 CMDE2:EV(CMDE.CENTERMANIFOLD) , PRINT("FLOW ON THE C.M. : "). PRINT(CMDE2»$

CENTER MANIFOLDS

35

Systems with Damping

Since the center manifold must contain a linearized system with eigenvalues equal to zero or having zero real part, it would appear that the method is inapplicable to the systems which have damping.

Carr [6] has shown.

however, that such cases can be handled by embedding the given system in a larger system which contains the damping as an additional dependent variable. For example. consider the previous example when damping is added:

(12)

x'

=~

x + y

(13)

y'

=-

x +

(14)

z'

=-

z + a x

~ y -

x z

2

Now to these equations we add the dummy equation:

o

(15)

The system (12)-(15) is now a 4-dimensional system with a 3-dimensional center manifold.

The damping terms

~

x and

terms in the neighborhood of the origin x

~

yare now nonlinear (quadratic)

= y = z = ~ = O.

results of the method are only valid for small values of

Naturally the

~.

In order to illustrate the procedure. we apply the program CM to this example:

CENTER MANIFOLDS

36 CM() ;

ENTER NO. OF EQS. 4; ENTER DIMENSION OF CENTER MANIFOLD 3; THE D.E. 'S MUST BE ARRANGED SO THAT THE FIRST 3 EQS. REPRESENT THE CENTER MANIFOLD. I.E. ALL ASSOCIATED EIGENVALUES ARE ZERO OR HAVE ZERO REAL PARTS. ENTER SYMBOL FOR VARIABLE NO. 1 MU; ENTER SYMBOL FOR VARIABLE NO. 2

X; ENTER SYMBOL FOR VARIABLE NO. 3

Y; ENTER SYMBOL FOR VARIABLE NO. 4 Z; ENTER ORDER OF TRUNCATION 3; ENTER RHS OF EQ. 1

D mu IDT = 0; ENTER RHS OF EQ. 2

D x IDT = MU*X+Y; ENTER RHS OF EQ. 3 DylDT= MU*Y-X-X*Z; ENTER RHS OF EQ. 4

D z IDT

=

-Z+ALPHA*XA 2;

dmu

--- = 0 dt

dx

-- = y

+ mu x

dt

dy

-- = -

x z + mu y - x

dt dz

-- = alpha

dt

2

x

- z

CENTER MANIFOLDS

37

CENTER MANIFOLD: 2 2 28 alpha mu y 2 alpha y - -------------- + ---------25 5

[z

8 alpha mu x y

2 alpha x y

+ -------------- - -----------

25

5 2

2

22 alpha mu x

3 alpha x + ----------]

25

5

FLOW ON THE C.M.:

2 dmu [--dt

dy

dx

= 0,

dt

=y

=-

+ mu x, dt

28 alpha mu y x (-

2 2 alpha y

-------------- + ---------25

5 2

8 alpha mu x y

2 alpha x y

22 alpha mu x

+ -------------- - ----------- - -------------- + 25

5

= 51

[VAX 8500 TIME

25

2

3 alpha x

----------)

+ mu y - x]

5

SEC.]

That is, for small values

of~,

the flow near the origin on the center

manifold of the system (12)-(15) is given by:

(16)

x'

=y

(17)

y'

= -x

+

~ x

3 x 2 + 5" 2 x y - 5" 2 y 2 + J1. [22 28 Y2] ) + ~ y + a x [- 5" 25 x 2 - 825 x Y + 25

Here the origin is unstable for is a Hopf bifurcation at

~

= o.

~

>0

and stable for J1.

< O.

Thus there

Let us use the Lindstedt method program LC2 of

Chapter 1 to approximate the size of the limit cycle as a function of the parameters a and /1..

In order to put the problem in a form suitable for

treatment by LC2. we scale x. y and J1. as follows:

(18)

x = EO

X.

Y= -

EO

y.

whereupon eqs. (16). (17) become:

CENTER MANIFOLDS

38

(19)

X'

(20)

Y,

- Y + M c

2

X

2 + 5' 2 X Y + 5' 22] M c 2 Y + a c2 X [3 5' X Y + •••

=X +

We continue with the application of the program LC2 to eqs.(19).(20):

LC2() ; The d.e.'s are of the form x'

-y + e*f. y'

=x

+ e*g

where f.g are of the form: quadratic + e*(linear + cubic) + e A 2*(quartic) + .

f

=e

mx

Enter g:

M*E*Y+ALPHA*E*X*(3*X 2/5+2*X*Y/5+2*Y 2/5); A

A

2

2

g

=

3x alpha e x (---- + ----- + ----) + e my 2 Y

2 x y

5

5

5

Enter truncation order: 2;

Done with step 1 of 2 Choices for limit cycle amplitude: 1)

0

2)

- 2 sqrt(5) sqrt(- -----) alpha

3)

2 sqrt(5) sqrt(- -----) alpha

m

m

Enter choice number 3; Done with step 2 of 2

CENTER MANIFOLDS m 2 sqrt(- -----) sqrt(5) cos(z) + . alpha

x

y

39

=2

m sin(z) sqrt( - -----) sqrt(5) + alpha 2

W

=1

-

11 e

m

-------

2

[VAX 8500 TIME

12 SEC. ]

That is. to lowest order approximation. the limit cycle amplitude is calculated as (cf. eq.(18»:

(21)

Eq.(21) asserts that for small have opposite signs.

~

the limit cycle exists only if

~

and a

As a check on this computation. we present in Fig.5 the

results of a numerical integration of the differential equations (12)-(14) for parameter values

~

= 0.01.

a

= -0.01.

As can be seen from the Figure. the

foregoing result based on center manifold and Lindstedt methods is in good agreement with the numerical computation.

CENTER MANIFOLDS

40

5

Y

x -5

5

-5 Fig.5. Numerical

(=

N) and center manifold/ Lindstedt

0(2) (= eM) solutions for the limit c)"cle of eqs. (12)-(14) for

~

=

0.01 and a

=

-0.01, displayed in the x-y plane.

CENTER MANIFOLDS

41

Eigencoordinates

The program CM requires that the coordinates on the center manifold be linearly uncoupled from the other coordinates of the problem.

In the case that

the original system is not in this form, some linear algebra needs to be done. We shall give an example pf this complication next, as we consider a bifurcation in the well-known Lorenz equations [14J:

(22)

x'

=a

(23)

y'

=p x

-

(24)

z'

=-

z + x y

(y - x)

f3

y -

x z

We shall be interested in the behavior of this system in the neighborhood of the equilibrium at the origin.

Linearizing about the origin,

we obtain

(25)

We use the ElGEN package in MACSYMA to compute the eigenvalues of the linearized system: (Here and in what follows we use the symbols s,r.b and e in MACSYMA to stand for a.p,t> and e respectively. linearized system (25) as A.)

A:MATRIX([-S,S,OJ,[R.-l.O].[O.O.-BJ); [ - s [ [ r [ [ 0

S

- 1

0

] ] 0 ] ] - b ] 0

We refer to the matrix of the

42

CENTER MANIFOLDS

EIGENVALUES(A) ; 2

sqrt(s

[[-

+ (4 r - 2) s + 1) + s + 1

---------------------------------2

2

sqrt(s

+ (4 r - 2) s + 1) - s ,- 1

---------------------------------- - b]. [1. 1. 1]] 2

Here the first list of three eigenvalues is followed by a second list of their respective multiplicities. is zero for all a manifold.

and~.

Thus at p

Note that at

=1

p

= lone

of the eigenvalues

we have a one-dimensional center

We propose to use our program eM to study the bifurcations of

equilibria in the neighborhood of the origin.

Before we can proceed. however.

we must uncouple the eigencoordinate corresponding to the center manifold from the other eigencoordinates.

We return to our previous MACSYMA session and

compute the eigenvectors of the linearized system when p = 1:

EIGENVECTORS(EV(A.R=I»; 1 [[[- s -

1. - b. 0]. [1. 1. 1]]. [1. - - 0]. [0. O. 1]. [1. 1. 0]] s

Here the first list contains the eigenvalues. the second their respective multiplicities. and the last three lists are their respective eigenvectors.

In order to diagonalize the linearized system. we transform

variables from x.y.z to u.v.w via a matrix whose columns are the preceding eigenvectors:

Now we need to substitute (26) into (22)-(24). MACSYMA utility program. TRANSFORM. to accomplish this.

We will use our own TRANSFORM performs an

CENTER MANIFOLDS

arbitrary coordinate transformation (not necessarily linear) and will be used again in later Chapters.

We follow the MACSYMA run with a listing of

TRANSFORM:

TRANSFORM ( ) ;

Enter number of equations 3;

Enter symbol for original variable 1

X; Enter symbol for original variable 2

Y; Enter symbol for original variable 3 Z;

Enter symbol for transformed variable 1

U; Enter symbol for transformed variable 2

V; Enter symbol for transformed variable 3 W; The RHS's of the d.e. 's are functions of the original variables: Enter RHS of x d.e. d x /dt =

S*(Y-X); d x /dt

=s

(y - x)

Enter RHS of y d.e. d y /dt =

R*X-Y-X*Z; d Y /dt

=-

x z - y + r x

Enter RHS of z d.e. d z /dt =

-B*Z+X*Y;

d z /dt

=x

Y- b z

The transformation is entered next: Enter x as a function of the new variables x =

U+V;

x

=v

+

U

Enter y as a function of the new variables y =

U-V/S; y

=u

v - -

s

43

CENTER MANIFOLDS

44

Enter z as a function of the new variables z =

W; z

=w du

5

(u

W

+ (1 - r) u) + 5 V (w - r + 1)

[[-- = - -------------------------------------, dt

5

+ 1

2

dv dt dw dt

s «r - 1) u - u w) + v (s (- w + r + 1) +

5

+ 1)

=- -------------------------------------------------5

5

2 2 (b w - u ) + v + (u -

+ 1

5

u)

V

= - -------------------------------]] s

[VAX 8500 TIME = 1 SEC.]

Here is the listing of TRANSFORM:

TRANSFORM():=( /* input data */

n:read("Enter number of equations"), for i:l thru n do x[i]:read("Enter symbol for original variable", i), for i:l thru n do y[i]:read("Enter symbol for transformed variable". i). print("The RHS's of the d.e. 's are functions of the original variables:"). for i:l thru n do ( print("Enter RHS of".x[i]."d.e."), print("d" .x[i], "/dt ="), f[i]:read() , print("d" .x[i]. "/dt =". f[i]». print("The transformation is entered next:"), for i:l thru n do ( print("Enter" ,x[i], "as a function of the new variables").

CENfER MANIFOLDS

45

print(x[i]. "="). g[i]: read(). print(x[ i]. "=" .g[i])). 1* do i t *1

for i:l thru n do depends([x[i].Y[i]].t). for i:l thru n do eq[i]:djff(x[i].t)=f[i]. trans:makelist(x[i]=gti].i.l.n). for i:l thru n do treq[i]:ev(eq[i].trans.diff). treqs:makelist(treq[i].i.l.n}. derivs:makelist(diff(y[i].t}.i.l.n). neweqs:solve(treqs.derivs»$

In order to observe the bifurcation of equilibria in the center manifold as

p

passes through unity. we set

(27)

p

1 + c

and we embed the system in a 4-dimensional phase space with c' = O.

The

following MACSYMA command substitutes (27) into the list NEWEQS which contains the results of TRANSFORM.

The right hand sides of the transformed equations

are stored in an array RHS. to be conveniently passed on to the program CM:

FOR 1:1 THRU 3 DO RHS[I]:EV(PART(NEWEQS.l.I.2).R=1+E); CM(} ; ENTER NO. OF

EQS.

4;

ENTER DIMENSION OF CENTER MANIFOLD 2;

THE D.E. 'S MUST BE ARRANGED SO THAT THE FIRST 2 EOS. REPRESENT THE CENTER MANIFOLD. I.E. ALL ASSOCIATED EIGENVALUES ARE ZERO OR HAVE ZERO REAL PARTS.

CENTER MANIFOLDS

46 ENTER SYMBOL FOR VARIABLE NO. 1 E;

ENTER SYMBOL FOR VARIABLE NO. 2

U; ENTER SYMBOL FOR VARIABLE NO. 3

V; ENTER SYMBOL FOR VARIABLE NO.4

W; ENTER ORDER OF TRUNCATION 2; ENTER RHS OF EQ. 1 D e IDT = 0;

ENTER RHS OF EQ. 2 D u /DT =

RHS[1];

ENTER RHS OF EQ. 3 Dv/DT= RHSt2] ; ENTER RHS OF EQ. 4 D w /DT

=

RHS[3]; de

-- = 0 dt

du

(u

8

W -

e u) + s

(w - e)

V

s + 1

dt

2

dv

s (e u - u w) + v

(8 (- W +

dt

8

2

dw dt

. 2

8

CENTER MANIFOLD: 2 e s u

[v

u

= - ------------, w = --] 2

8

+ 1

(b w - u ) + v + (u -

8

b

+25+1

e + 2) +

8

u)

V

5

+ 1)

CENfER MANIFOLDS

47

FLOW ON THE C.M.: 2 2 e s

3

u (-- - e) b

u

s (-- - e u) b

de

du

dt

dt

[-- = 0,

u

2 + 2 s + 1

s

- ------------------------------J s + 1

[VAX 8500 TIME

13 SEC. J

The resulting expression from CM, stored in the variable CMDE2, can be cleaned up by using the MACSYMA function FACTOR:

FACTOR(CMDE2) ; 2

de

du

dt

dt

[-- = 0,

s (s

2

- e s + 2 s + 1) u (u

- b e)

-----------------------------------J 3 b (s + 1)

The last equation gives an approximation for the flow on the center manifold.

There are 1 or 3 equilibria, depending on the sign of the product

13 e: (2B)

o

u

and

=+

u

(13 e)1/2

Thus there is a pitchfork bifurcation at p

= 1,

see Fig.6.

In order to check the center manifold computation, we calculated the center manifold numerically.

It is straightforward to do this since any

initial condition close to the origin produces a motion which asymptotically approaches the center manifold.

In Fig.7 we plot in the x-y plane the

numerical results for Lorenz's original parameter values of a and for p

= 0.9.

= 10,

f3

= 8/3,

Fig.7 also shows the results of the preceding analysis,

obtained as follows:

We invert the change of coordinates (26) and transform

the center manifold equations:

CENTER MANIFOLDS

48

u

Fig.G. Pitchfork bifurcation of equilibria on the center manifold at the origin in the Lorenz equations, see eq.(28). 1

y CM

1

Fig.7. Numerical

(=

N) and center manifold

(=

x

CM) solutions

for the center manifold at the origin in the Lorenz equations for a

= 10, B = 8/3, P = 0.9,

~

= -0.1, displayed in x-y plane.

CENTER MANIFOLDS (29)

v

au

E.

- (1+a)2

2

u

w

49

i3

giving 2

(30)

For a

y

= 10,

f3

(a+l) + (a+1)2 -

8/3 and

E.

z

X

a

E.

~

-0.1 this projects onto the x-y plane as the

straight line

(31)

0.99 ... x

y

Exercise

1. This problem concerns the behavior of van der Pol's equation at infinity [19].

If in van der Pol's equation,

(PI)

w

+ w-

v

w' / w

2 (1 - w ) w'

E.

o

we set [27]:

(P2)

and

and reparameterize time with dT

= w2

(P3.1)

v

- E.

+

(P3.2)

z

- v z

V

Z

2

z

1 / w

dt, we obtain:

(E.

V -

2

v-I)

3

where primes now represent derivatives with respect to T. The equilibrium at infinity, v

= z = 0,

has eigenvalues zero and

There is a center manifold tangent to the z-axis. . . f an approxImatIon or·It to 0(z12).

-E..

Use the program CM to obtain

Note that unlike the treatment of van der

Pol's equation by Lindstedt's method, we do not assume

E.

is small here.

CHAPTER 3

NORMAL FORMS

Introduction

Like Lindstedt's method, the method of normal forms is used for obtaining approximate solutions to ordinary differential equations.

In

contrast to Lindstedt's method, however, the method of normal forms does not involve expanding the solution in an infinite series.

Rather, the idea is to

transform the differential equations themselves into a form which is easily solved.

The method involves generating a transformation of coordinates (i.e.

dependent variables) in the form of an infinite series, and computing the coefficients of the series so that the resulting transformed differential equations are in a normal (i.e., a simple or canonical) form. The method may be illustrated by considering a system of two differential equations (although the same process applies to an n dimensional system) : (1. 1)

x'

f(x,y)

(1.2)

y'

= g(x,y)

We assume that the coordinates x and y have been chosen so that the origin x

=y =0

is an equilibrium point:

NORMAL FORMS f(O,O)

(2)

g(a,a)

51

a

We Taylor expand f and g about the origin,

(3.1)

x'

=a

x + b y + F(x,y)

(3.2)

y'

=c

x + d y + G(x,y)

where F and G are strictly nonlinear (i.e., contain terms of degree 2 and higher) . In the case that F and G contain quadratic terms, we transform from x,y to u,v coordinates via a near-identity transformation with general quadratic terms: (4.1)

x

(4.2)

y

=u

+ A120 u

2

v + A220 u

2

+ AlII u v + Al02 v

+ A211 u v +

~02

v

2

2

where the A. are to be determined. 1

We substitute eqs.(4) into eqs.(3) and neglect terms of order 3 (which do not influence the quadratic coefficients Ai). linear in the derivatives u' and v'. u

= v = 0,

The resulting equations are

We solve for these and again expand about

neglecting terms of degree 3 and higher:

(5.1)

u'

a u + b v + C u 120

(5.2)

v

c u + d v + C u 220

2

+ C

u v + C

+ C

u v + C v 202

lll

2

2ll

l02

v

2

2

where the coefficients C. are known linear functions of the A.. 1

1

The linear

terms in (5) are identical to the linear terms in (3) due to the near-identity

NORMAL FORMS

52

nature of the transformation (4). At this point the coefficients A. are chosen so as to put eqs.(5) into 1

a normal form.

If the equilibrium point is hyperbolic (i.e. if all the

eigenvalues have nonzero real parts), then we can always remove all nonlinear terms [14,§3.3].

However, in some problems (involving resonances or repeated

zero eigenvalues) this is not possible.

In such cases the choice of the normal

form is somewhat arbitrary [11]. Once the coefficients in (4) have been determined, we may extend the transformation from x,y to u,v coordinates to include cubic terms.

(Note that

even if eqs.(3) do not contain cubic terms, the transformation (4) will generally introduce cubic terms.)

Proceeding as before we compute the new

equations on u and v neglecting terms of order 4, and choose the coefficients of the cubic terms in the near-identity transformation in order to best simplify the resulting differential equations.

Note that the transformation up

to and including cubic terms will not affect the already determined quadratic terms in the normal form. Proceeding in this fashion we can (in principle) generate the desired transformation to any order of accuracy.

Note, however, that the use of

truncated power series can be expected to restrict the applicability of the method to a neighborhood of the origin.

NORMAL FORMS

53

The Case of a Pair of Imaginary Eigenvalues

As our first example of the method we will consider the system

(6.1)

~'

(6.2)

y'

=-

y +

~

F(~.y)

+ G(x,y)

where F apd G are strictly nonlinear in x,y.

The problem here is to determine

the stability of the equilibrium at the origin.

~ineari~tion about

the origin

is no help, since the origin is a center (cf. Hartman's theorem [14]). Takens [45]

~s ~hown

that this important class of systems can be put

in the normal form:

(7.1)

r

= Bl

(7.2)

", o

=1

r3 + B

2

r5 + B

3

r 7 + •••

D r6 + + D1 r 2 + D2 r 4 +'3

where r and 6 are polar coordinates

u

(8)

=r

cos 6,

v

=r

sin 6

and where u,v are related tq x,y by a near-identity transformation. rectangular coordinates, eqs.(7) become:

(9.1)

u

(9.2)

v'

- v + B (u2 + 1

y

2

)

U -

D (u2 + v 2 ) v + 0(5) 1

2 2 2 2 u + B (u + v ) v + D1 (u + v ) u + 0(5) 1

In

54

NORMAL FORMS

Comparison of eqs.(9) and (5) shows that all the quadratic terms can be eliminated in this case. although terms of odd power will in general remain. We shall present several MACSYMA programs which will be used to treat this problem [38J. detail.

Before presenting a sample run. we discuss a computational

If the original equations are written in the form

x·

(10)

f{x)

where x and f(x) are n-vectors, cf. eqs.(I). then the application of a near-identity transformation,

(11)

x

=u

+ g(u)

where u and g(u) are n-vectors, cf. eqs.(4). gives:

(12)

(I + Dg) u'

feu + g(u»

where Dg is the n x n Jacobian matrix.

(13)

u'

= (I

+ Dg)

-1

Eq.(12) may be solved for u' as

feu + g(u»

Now this matrix inversion may be accomplished approximately by using the expansion:

(14)

(I + Dg)-1

=I

_ Dg + (Dg)2 _ (Dg)3 + ••• + (_Dg)m-l

where m is the degree of the highest order terms retained in (13) truncation.

~fter

(The series (14) is truncated at degree m-l terms since it is

multiplied by f(u+g(u»

which begins with linear terms.)

We use this scheme in

NORMAL FORMS the computer algebra program NF which follows.

55

We found it to be faster than

using the MACSYMA SOLVE routine to isolate the derivatives u' in eq.(12). We return now to eqs.(6) which we write in the truncated form:

x

(15.1)

2 2 3 2 x Y L+ F ~+ F ~+ F x y + F - y + F xy xx2 yy2 xxx 6 xxy -22

F

xyy

2

3 F

L+

yyy 6

2 2 3 2 x Y ~+ G x + G ~+ G x y + G L+ G -2xy xx2 yy2 xxy xxx 6

y

(15.2)

~+

2

3

~+ G L+ xyy 2 yyy 6

G

The MACSYMA run which applies normal forms to this system consists of the functions NF. GEN. DECOMPOSE. HOPF2. HOPF3, and a function used in Chapter 2. TRANSFORM.

We will annotate the run in Italics (to distinguish the

comments from the run itself) and will follow it with the program listings:

NF()$ DO YOU WANT TO ENTER NEW VARIABLE NAMES (YIN)?

Y; HOW MANY EQS 2;

SYMBOL FOR OLD X[

]

X; SYMBOL FOR OLD X[ 2 ]

Y; SYMBOL FOR NEW X[ 1 ] U; SYMBOL FOR NEW X[ 2 ] V; DO YOU WANT TO ENTER NEW D.E.·S (YIN)?

Y;

56

NORMAL FORMS

ENTER RHS OF EQ. NO. 1 .n x IDT = -Y+Fxx/2*X"2+FXY*X*Y+FYY/2*Y"2+FXXXl6*X"3+FXXY/2*X"2*Y+FXYY/2*X*Y"2+FYYY/6*Y"3; 3 fyyy y

n x IDT =-------

2 fxyy x y

2

2

3

fxxY x . y

fyy y

fxxx x

+ --------- + ------ + --------- + fxy x y - y + ------622 2 6 2 fxx x + -----2

ENTER RHS OF EQ. NO. 2 .n y IDT = X+GXXI2*X"2+GXY*X*Y+GYY/2*Y"2+GXXX16*X"3+GXXY/2*X"2*Y+GXYY/2*X*Y"2+GYYY/6*Y"3; 3 gyyy y

n y /DT

= -------

2 gxyy x y

2

2

3

2

y gxxy x y gxxx x gxx x + --------- + ------ + --------- + gxy x Y + ------- + -----gyy

622

2

6

2

+x INPUT NEAR-IDENTITY TRANSFORMATION (USE PREV FOR PREVIOUS TRANSFORMATION)

=u

x

+ ?

We wish to enter a quadratic neqr-identity coeFFicients, as in eq.(4).

transFormatio~ with

The program GEN is designed to

Frequently needed step.

GEN(2) ; 2 v +a

x=a 1. [0. 2]

2 u +u

u v +a 1.

[1. 1]

I, [2. 0]

y = v + ? GEN(2);

2 2 uv+v+a u v +a y=a 2, [2. 0] 2. [1, 1] 2. [0. 2] ENTER TRUNCATION ORDER (HIGHEST ORDER TERMS TO BE KEPT) 2;

2

du

-- =-

dt

+ 2 a - fxx) u v - «2 a 2. [2. 0] 1. [I, 1] - 4 a

+ (2 a

1. [2. 0]

2. [1. 1]

+ 4

- 2 fxy) v u

~

1. [0. 2]

2 - fyy) v )/2 +

- 2 a

+ (2 a

2, [0. 2]

1. [ 1. 1]

general

F~cilitate

this

NORMAL FORMS dv

-- = u -

«2 a

dt

57

2 - 2 a

1. [2. 0]

2. [1. 1]

+ (- 4 a

- gxx) u

+ (- 2 a

- 2 gxy) v u

- 2 a

+4a

2, [0, 2]

2. [2, 0]

1, [1. 1]

2 - gyy) v )/2 +

- 2 a

1. [0, 2]

2, [1. 1]

DO YOU WANT TO ENTER ANOTHER TRANSFORMATION (Y/N)

N; [VAX 8500 TIME

=9

SEC.]

These last equations correspond to eqs.(5) in our general treatment.

We must

now isolate the coefficients C. (see eqs.(5». a frequently needed step which t

is facilitated by the utility program DECOMPOSE.

Then we wish to choose the

coefficients a.t, [.J ,I~ L] so that the transformed equations are in the normal

The program HOPF2 is designed to automate these steps:

form (9).

HOPF20;

2 gyy + gxx + 2 fxy [[a 1, [2. 0]

6

gxy - fyy + fxx

- 2 gxy + 2 fyy + fxx

a

a

1. [1, 1]

6

2. [2, 0]

gyy - gxx + fxy

- gyy - 2 gxx + 2 fxy a

a 2, [1, 1]

a

3

3

1, [0, 2]

6

2 gxy + fyy + 2 fxx = -------------------]] 2. [0. 2] 6

[VAX 8500 TIME

=6

SEC.]

Having obtained the unknown quadratic coefficents ai.[j.k]' we return to NF again to check the preceding work and to extend it to cubic terms:

NORMAL FORMS

58

NF()$ DO YOU WANT TO ENTER NEW VARIABLE NAMES (YIN)?

N; DO YOU WANT TO ENTER NEW D.E. 'S (YIN)?

N; INPUT NEAR-IDENTITY TRANSFORMATION (USE PREY FOR PREVIOUS TRANSFORMATION) x =u + ?

The results of the previous step (which may be referred to by

%) may be

substituted into the previous transformation (called PREY here) as follows:

EV(PREV,%); 2 (- gyy - 2 gxx + 2 fxy) v

x

=--------------------------

(gxy - fyy + fxx) u v + ---------------------

6

3

2

(2 gyy + gxx + 2 fxy) u - ------------------------ + u 6

=

y v + ? EV(PREV,%) ; 2 y

(2 gxy + fyy + 2 fxx) v

(gyy - gxx + fxy) u v

6

3

= ------------------------ - ---------------------

+ v

2 (- 2 gxy + 2 fyy + fxx) u + -------------------------6

ENTER TRUNCATION ORDER (HIGHESf ORDER TERMS TO BE KEPT) 2; du

-- = -

v +

dt dv = u +

dt

This confirms that the routine is working so far. terms to the previous transformation:

Now we add general cubic

NORMAL FORMS

59

00 YOU WANT TO ENlER ANOTIIER TRANSFORMATION (YIN)

Y; INPUT NEAR-IDENTITY TRANSFORMATION (USE PREY FOR PREVIOUS TRANSFORMATION) x u + ? PREV+GEN(3) ;

=

2 3

=a

x

+ a

v

1. [0. 3J

1. [~. 2J

+ -------------------------6

U V

(gxy - fyy + fxx) u v 3 v + --------------------- + a u 3 1. [3. OJ

2 + a

(- gyy - 2 gxx + 2 fxy) v

2

u

1, [2. 1J

2 (2 gyy + gxx + 2 fxy) u - ------------------------ + u 6

y

=v

+ ?

PREV+GEN(3) ;

2

=a

y

3 v

+ a

2 u

2. [2. 1J

u v

2. [1, 2J

2. [0. 3J

+ a

(2 gxy + fyy + 2 fxx) v

2

+ -----------------------6

(gyy - gxx + fxy) u v 3 v - --------------------- + v + a u 3 2. [3. OJ

2 (- 2 gxy + 2 fyy + fxx) u + -------------------------6 ENTER TRUNCATION ORDER (HIGHEST ORDER TERMS TO BE KEPT)

3;

du

-- = -

v -

«

(fxy + gxx + 2 gyy) fxx - 2 fxy fyy + 2 fxy gxy + 6 a

dt

2. [3. OJ + 6 additionat tines omitted here for brevity

dv

2

-- = u

+ «gxy fxx + 2 gxy fyy - 2 gxy

2 - 2 gxx fxy - gxx - 2 gyy gxx

dt + 7 additionat tines omitted here for brevity 00 YOU WANT TO ENTER ANOTHER TRANSFORMATION (YIN)

N;

[VAX

8500 TIME

= 47

SEC.J

60

NORMAL FORMS

Next we cat!. HOPF3 which DECOMPOSEs the last equations and selects the cubic

a

coeFFicients such that we obtain the normal Form (9).

i.U.k.]

Note that this

involves 6 linear algebraic equations in 8 unknowns and thus the general solution contains two arbitrary constants.

Happily the MACSYMA Function SOLVE

automatically does the linear algebra For us. and assigns the arbitrary constants the values %r1 and %r2: HOPF3();

= (2

[[a

2

2

+ (gxx + 7 fxy) gyy - 3 gxyy - 2 gxy

gyy

1, [3. 0]

2 2 + (- fyy - 7 fxx) gxy - gxxx + 3 gxx + fxy gxx - fyyy - 3 fyy - fxx fyy 2 + 2 fxy

2 - 3 fxxy - 2 fxx

= %r2.

+ 24 %r1)/24. a

2. [3. 0] + 13 additional lines giving the other ai.[j.k.]·s in terms of the given

constants of the problem and the two arbitrary constants %r1 and %r2. [VAX 8500 TIME = 34 SEC.]

Next we substitute these values of the a. [. LJ'S into the transFormed 1..

J ,,''-

diFFerential equations. stored in the variable NEWDES by the program NF: EXPAND(EV(NEWDFS.%)) ; 2 3 gyy v

du

[-dt

= -------

3

3

5 gxx gyy v +

------------

24

5 fxy gyy v -

------------

48

2

3

-

48

gxyy v

gxy

------- + 16

3

v

-------

24

+ 48 additional terms. none of which contain either of the arbitrary

constants %r1 or %r2 3 dv dt

gyyy v

= ------- -

3

gxy gyy v

---------- + ---------16

16

3

3

fyy gyy v

16

gxx gxy v

- ---------16

3

gxxy v

+ -------

16

+ 47 additional terms with no %r1 or %r2 dependence.

[VAX 8500 TIME

= 14

SEC.]

3

fxx gxx v

- ---------16

NORMAL FORMS

61

Note that although the transformation to normal form is not unique (since %rl and %r2 are arbitrary). the normal form itself does not depend on %rl or %r2.

Next we use the function TRANSFORM given in Chapter 2 to convert the transformed differential equations to the polar form (7):

TRANSFORM() ;

Enter number of equations' 2: Enter symbol for original variable 1

U; Enter symbol for original variable 2

V; Enter symbol for transformed variable 1 ~;

Enter symbol for transformed variable 2 THETA; The RHS's of the d.e. 's are functions of the original variables: Enter RHS of u d.e. d u /dt =

We specify the previously obtained differential equations by using the MACSYMA . sympol % :

RHS(PART(%. 1»: Enter RHS of v d.e. d v /dt = RHS(PART(%,2» ; The transformation is entered next: Enter u as a function of the new variables

= = v = RMSIN(THETA) ; v = r sin(theta)

u

R*COS(THETA) ; u r cos(theta) Enter v as a function of the new variables

dr

[[-- = «gyyy

+ (fyy - gxy) gyy - gxx gxy + gxxy - fxx gxx + fxy fyy

dt

3 + fxyy + fxx fxy + fxxx) r

2 sin (theta)

+ (gyyy + (fyy - gxy) gyy - gxx gxy + gxxy - fxx gxx + fxy fyy + fxyy 3 + fxx fxy + fxxx) r

2 cos (theta»/16.

NORMAL FORMS

62

2

dtheta «2 gyy

2 + (5 gxx - 5 fxy) gyy - 3 gxyy + 2 gxy

dt + 5 additionaL Lines omitted for brevity.

[VAX 8500 TIME = 21 SEC.]

This poLar form can be simpLified by using the MACSYMA function TRIGSIMP:

TRIGSIMP(%) ;

dr

[[-- = (gyyy

+ (fyy - gxy) gyy - gxx gxy + gxxy - fxx gxx + fxy fyy

dt 3 dtheta + fxyy + fxx fxy + fxxx) r 116. dt

2 - ((2 gyy

2 + (5 gxx - 5 fxy) gyy - 3 gxyy + 2 gxy 2

- 3 gxxx + 5 gxx

2 + 2 fxx ) r

+ (- fyy - 5 fxx) gxy

2 - fxy gxx + 3 fyyy + 5 fyy

2

+ 5 fxx fyy + 2 fxy

+ 3 fxxy

2 -

48)/48]]

[VAX 8500 TIME = 24 SEC.]

Here is the program listing:

1* THIS FILE CXlNTAINS NF(). A NORMAL FORM UTILITY FUNCTION. IT ALSO CXlNTAINS THESE ADDITIONAL FUNCTIONS: GEN(N) WILL GENERATE A HOMOGENEOUS ORDER N TRANSFORMATION. DECXlMPOSE() ISOLATES

THE

CXlEFFICIENTS OF THE NEW EQUATIONS.

VARS(N) GENERATES A LIST OF UNKNOWN CXlEFFICIENTS OF DEGREE N. HOPFk(), FOR k=2.3,4,5,6.7 SOLVES FOR THE CXlEFFICIENTS OF A SYSTEM OF 2 DE'S SO AS TO PUT THE EQS IN HOPF NORMAL FORM *1

NORMAL FORMS

NF(): = BLOCK( 1* NEW VARIABLE NAMES? *1

TEST : READ ("00 YOU WANT TO ENTER NEW VARIABLE NAMES (YIN)?"), IF TEST = N THEN GO(JUMP), N : READ ("HOW MANY EQS"), FOR I THRU N 00 (X[ I J

READ ("SYMBOL FOR OLD X[", I , "]") ) ,

FOR I THRU N OO( Y[IJ

READ ("SYMBOL FOR NEW X[",I,"]"»,

FOR I THRU N 00 DEPENDS([X[IJ,Y[IJJ,T), KILL(FLAG), 1* FLAG USED IN GEN *1

JUMP, 1* NEW D.E. 'S? *1

PRINT ("00 YOU WANT TO ENTER NEW D.E.'S (YIN)?"), TEST: READ () , IF TEST = N THEN GO(LooP), FOR I THRU N 00 (RHS[IJ:READ("ENTER RHS OF EQ. NO.",I,",D",X[IJ,"IDT ="), PRINT("D",X[IJ,"IDT =",RHS[IJ», KILL(RHS2) , RHS2[I,JJ := RHS[IJ, RHS3:GENMATRIX(RHS2,N,1),

LOOP, 1* NEAR-IDENTITY TRANSFORMATION *1

PRINT("INPUT NEAR-IDENTITY TRANSFORMATION (USE PREV FOR PREVIOUS TRANSFORMATION)"), FOR I THRU N 00 (ROW: I, PREV :TR[IJ,

64

NORMAL FORMS

TR[I] :READ (X[I],"=",Y[I]."+ ?"), PRINT (X[I]."=",Y[I]+TR[I]», 1* INPUT TRUNCATION ORDER *1

TRANS : MAKELIST (X[I]=Y[I]+TR[I],I.l.N), M : READ("ENTER TRUNCATION ORDER (HIGHEST ORDER TERMS TO BE KEPT)"). 1* TRANSFORM THE D. E. 'S *1

TEMP2 :EV{RHS3. TRANS), 1* SOLVE FOR THE TRANSFORMED DERIVATIVES *1

KILL{ JACOB) . JACOB[I.J]:=DIFF{TR[I].Y[J]). JACDB2:GENMATRIX{JACOB.N ,N), TEMP3:SUM{{-1)~I*JAODB2~~I.I.O.M-1).TEMP2.

1* TAYLOR EXPAND THE RESULTING EOS *1

NEWRHS : TAYLOR(TEMP3,MAKELIST{Y[Ij.I.l.N).O.M). NEWDES:MAKELIST{DIFF{Y[I],T)=NEWRHS[I.l].I.l.N). FOR I: 1 TItRu N 00 PRINT{PART{NEWDES.I». 1* ENTER ANOTHER TRANSFORMATION? *1

BRANCH:READ{"OO YOU WANT TO ENTER ANOTHER TRANSFORMATION (YIN)"). IF BRANCH = Y THEN GO{LOOP). NEWDES) $

1* AUXILIARY FUNCTIONS *1

GEN{NN):={ IF NOT NUMBERP{FLAG[NN]) THEN ( SUB:MAKELIST{K[I].I,l,N), VAR:PRODUCT(Y[I]~K[I],I,l,N),

TEMPGEN: A[ROWDUMMY ,SUB]*VAR. FOR I: 1 THRU N 00

NORMAL FORMS

TEMPGEN:SUM(EV(TEMPGEN.K[I]=J).J.O.NN). TEMPGEN2:LAST(TAYLOR(TEMPGEN.MAKELIST(Y[I].I.I.N).O.NN)). TEMPGEN3[NN] :EXPAND(TEMPGEN2) . FLAG[NN] : I). EV(TEMPGEN3[NN]. ROWDUMMY=ROW)) $

DECXlMPOSE():=( KILL(C) . IF NOT NUMBERP(FLAG[M]) THEN GEN(M). TEMP8:SUBST( .. [ ..... + ... TEMPGEN3[M]). TERMS:EV(TEMP8.A[DUMMY.SUB]:=I). CXlEFFS:EV(TEMP8.A[DUMMY.SUB]:=e[DUMMY.SUB].MAKELIST(Y[I]=I.I.I.N)). FOR I: I THRU N DO( FOR J: I THRU LENGTH(TERMS) DO( EV(PART(CXlEFFS.J) .ROWDUMMY=I) :: RATCXlEF(EXPAND(NEWRHS[I.I]).PART(TERMS.J)))))$

VARS(NN):=( TEMP5:SUM(EV(TEMPGEN3[NN]) .ROWDUMMY .I.N). TEMP6: SUBST (.. [ ..... +" •TEMP5) . TEMP7:EV(TEMP6.MAKELIST(Y[I]=I.I.I.N)))$

HOPF2():=(DECXlMPOSE(). SOLVE([C[1.[2.0]].C[1.[1.1]].C[1.[O.2]].C[2.[2.0]]. C[2.[1.1]].C[2.[O.2]]]. VARS(2)))$

NORMAL FORMS

66

HOPF3{):={DECOMPOSE(),

SOLVE([C[1,[3,0]]=C[1,[1,2]],C[1,[3,O]]=C[2,[2,1]], C[1,[3,O]]=C[2,[O,3]],C[1,[0,3]]=C[1,[2,1]], C[1,[O,3]]=-G[2,[3,0]],C[1,[0,3]]=-C[2,[1,2]]], VARS(3»)$

HOPF4{):={DECOMPOSE(),

SOLVE{[C[1,[4,O]],C[1,[3,1]],C[1,[2,2]].C[1,[1,3]], C[1.[0,4]],C[2,[4.0]].C[2.[3.1]],C[2.[2.2]], C[2.[1.3]].C[2,[O,4]]], VARS(4»)$

HOPF5{):={DECOMPOSE().

SOLVE{[C[1,[5.0]]=C[1.[3.2]]/2.C[1.[5.0]]=C[1.[1.4]]. C[1.[5,O]]=C[2.[4.1]],C[1.[5,O]]=C[2.[2.3]]/2, C[1,[5.0]]=C[2,[0.5]].C[2, [5.0]]=C[2, [3.2]]/2. C[2.[5,O]]=C[2.[1.4]],C[2.[5.0]]=-C[1.[4.1]], C[2. [5.0]]=-G[1. [2,3]]/2.C[2. [5.0]]=-C[1. [0,5]]]. VARS(5»)$

HOPF6():=(DECOMPOSE().

SOLVE{[C[1.[6,O]].C[1.[5.1]].C[1.[4.2]],C[1.[3.3]], C[1.[2.4]].C[1.[1.5]].C[1.[0.6]],C[2.[6.0]]. C[2.[5,1]].C[2.[4,2]],C[2.[3,3]],C[2.[2,4]]. C[2.[1.5]],C[2.[0.6]]]. VARS(6»)$

NORMAL FORMS

67

HOPF7():=(DECOMPOSE(). SOLVE([C[1.[7.0]]=C[1.[5.2]]/3.C[1.[7.0]]=C[1.[3.4]]/3. C[1.[7.0]]=C[1.[1.6]].C[1.[7.0]]=C[2.[6.1]]. C[1.[7.0]]=C[2.[4.3]]/3.C[1.[7,O]]=C[2.[2.5]]/3. C[1.[7.0]]=C[2.[O.7]].C[2.[7,O]]=C[2.[5.2]]/3, C[2,[7.0]]=C[2.[3.4]]/3.C[2.[7.0]]=C[2.[1.6]]. C[2.[7.0]]=-C[1.[6.1]].C[2.[7.0]]=-C[1.[4.3]]/3. C[2.[7.0]]=-C[1,[2.5]]/3,C[2.[7.0]]=-C[1.[O.7]]], VARS(7»)$

Thus we have shown that the problem specified by eqs.(15) can be transformed to the normal form:

(16)

S'

=1

+ D r

1

2 + ••

where (17)

16 B1 = Gyyy + Gxxy + Fxyy + Fxxx + Fyy Gyy - Fxx Gxx

-Gxx Gxy -Gyy Gxy +Fxx Fxy +Fyy Fxy

and where D is known. 1 The significance of this calculation is now clear: the origin is given by the sign of B . From (16). B > 0 1 1 makes the origin r = 0 unstable (respectively stable).

The stability of (respectively

< 0)

Note that this problem could not have been handled by Lindstedt's method. which gives no information regarding stability. As an application of these results, we may now resolve an unfinished question raised in connection with eqs.(1)-(11) in Chapter 2. that the stability of the origin in the system:

We showed there

68

NORMAL FORMS

(18.1)

x'

(18.2)

y

(18.3)

z

=y - x - x z

=-

z + a x

2

could be reduced via center manifold theory to the simpler question of the stability of the origin in the system:

(19.1)

x

=y

(19.2)

y'

= - x -

3 "5

a x

3

2 +"5 a

x

2

y -

2 "5

a x y

2

+

This system will be in the form of eqs.(6) if we interchange x and y. Computation of the quantity B of eq.(17) gives: 1

(20)

Thus the origin in the center manifold flow (19). and hence in the original three-dimensional flow (18). is stable if a

<0

and unstable if a

See Fig.8 where this behavior is confirmed by numerical integration of the system (18).

> O.

NORMAL FORMS

2

y

0.1

CI.

x -2

2

B

-2

2

y CI.

=

-0.1

x

-2

2

-2 Fig.S. Numerical solutions of the system (18) displayed in the x-y plane for

CI.

= 0.1 and -0.1.

In both cases,

A corresponds to the initial condition t=O,x=l,y=O,z=O, while B corresponds to the system at time t=50.

NORMAL FORMS

70

Before leaving the system (6), we note that by including damping in (6) we may use normal forms to study the Hopf bifurcation.

(21.1)

x'

- y +

~

x + F(x,y)

(21.2)

y'

x +

~

y + G(x,y)

The system

is best studied by embedding it in a three-dimensional system with variables x,y

and~,

i.e., by appending to it the dummy equation:

o

(21.3)

This approach permits the

~

x and

~

y terms to be considered as

nonlinear in the neighborhood of the origin x

= y = ~ = O.

By applying normal forms to this system, we obtain the polar result:

(22)

where B

(23)

r

1

=~

r + B

1

is given by (17).

r

r

3 + •••

e'

1 + •••

The r equation has an equilibrium at

(_ ~ / B )1/2 1

this being the radius of the limit cycle.

Note that this result agrees with

that derived by Lindstedt's method in Chapter I, eqs.(26),(27). Having derived this result by both Lindstedt's method and normal forms, a comparison between the two methods can be made.

Most significant is that

normal forms provides a dynamic on the process (eq.(22)), while Lindstedt's method offers only the limit'cycle radius.

Moreover, Lindstedt's method

requires the inclusion of a small perturbation parameter c in the equations,

71

NORMAL FORMS

while normal forms does not.

On the other hand, all results of the method of

normal forms are restricted to a neighborhood of the origin, while Lindstedt's method (which is valid in the neighborhood of the Hamitonian system

= 0)

corresponding to c

can be used to approximate limit cycles with large

amplitude, i.e. which do not necessarily lie in a neighborhood of the origin. 1~)

(See, however, Exercise

The Case of a Double Zero Eigenvalue

In order to motivate our next example of the method of normal forms, we will investigate the behavior of van der Pol's equation at infinity [19J,[39J. If in van der Pol's equation,

(24)

w"

2 + w - c (1 - w ) w'

x

w / w'

o

we set [27J:

(25)

and reparameterize time with dT

(26.1)

x

z

2

(1 + x

and

z

= w,2 2

~

1 / w'

dt, we obtain:

c x) + c x

3

(26.2)

where primes now represent derivatives with respect to T. As w'

~

00

,

z

~

line in the w,w' plane.

0, from (25).

Moreover x is constant along a radial

Thus as we move outward to infinity along such a line,

z approaches 0, and we may think of z

=0

as the "line at infinity"

NORMAL FORMS

72

parameterized by x.

Note that z

0 satisfies eq.(26.2). and thus is an

integral curve for this system. Substituting z = 0 into (26.1) shows that x at infinity for the system (26).

= z =0

is an equilibrium

We are interested in the nature of the local

behavior in the neighborhood of this point. Note that the system (26) has no linear part.

In order to simplify the

analysis, we set

(27)

y = z

2

whereupon (26) becomes:

,

(28.1)

x

(28.2)

y'

.

3

=y-exy+eX

=

2

+x

y

222 -26Y +2xy +2ex y

The problem here is to determine the stability of the equilibrium at the origin in eqs.(28).

Note that linearization about the origin is no help

since (28) has a double zero eigenvalue at the origin (cf. Hartman's theorem. [14]).

System (28) is a particular case of the system:

(29.1)

X·

=y

(29.2)

y'

=

+ F(x,y)

G(x.y)

where F and G are strictly nonlinear in x.y.

Takens [45] has shown that this

class of systems can be put in the normal form:

NORMAL FORMS

73

(30.1 )

(30.2)

v'

where u and v are related to x and y by a near-identity transformation. We note that Bogdanov [5J and Guckenheimer and Holmes [14J have chosen the alternate normal form for this problem:

(31.1 )

u' = V

(31.2)

V'

which is obtainable from (30) via the near-identity transformation:

(32)

u = U

and

v = V-

2 bn un

n=2

We write eqs.(29) in the truncated form:

(33.1)

x'

=y

2

+ F

~ + F

xx2

xy

x y + F

2 L + F

3

2

~ + F ~ xxx 6 xxy 2

yy2

2

3

~+F L+ xyy 2 yyy 6

+F

(33.2)

y'

=

2

G

x xx 2

2

+G

xy

xy+G

3 ~+G

L+G

xxx 6

yy 2

2

+G

xyy

xxy

3 L+ yyy 6

~+G

2

2

~

2

74

NORMAL FORMS

The following MACSYMA run applies our normal form utilities to this problem.

Our goal is to choose a near-identity transformation which puts the

system (33) into the normal form (30):

NF()$ DO YOU WANT TO ENTER NEW VARIABLE NAMES (YIN)?

Y; HOW MANY EQS

2; SYMBOL FOR OLD X[ 1 ]

X; SYMBOL FOR OLD X[ 2 ]

Y; SYMBOL FOR NEW X[ 1 ]

U; SYMBOL FOR NEW X[ 2 ]

V; DO YOU WANT TO ENTER NEW D.E.'S (YIN)?

Y; ENTER RHS OF EQ. NO. 1 ,D x IDT

=

Y+FXX*XA2/2+FXY*X*Y+FYY*YA2/2+FXXX*XA3/6+FXXY*XA2*Y/2+FXYY*X*YA2/2+FYYY*Y A3/6; 3 fyyy y

D x /DT

= ------6

2 2 2 3 fxyy x y fyy y fxxy x y fxxx x + --------- + ------ + --------- + fxy x y + y + ------2 2 2 6 2 fxx x + -----2

ENTER RHS OF EQ. NO. 2 .D y /DT

=

GXX*XA2/2+GXY*X*Y+GYY*YA2/2+GXXX*XA3/6+GXXY*XA2*Y/2+GXYY*X*YA2/2+GYYY*Y A3/6; 322 gxyy x y gyy y

gyyy y

D y /DT

= ------6

3

2 gxxy x

2

gxx x + --------- + ------ + --------- + gxy x y + ------- + -----2 2 2 6 2 y

gxxx x

INPUT NEAR- IDENTITY TRANSFORMATION (USE PREV FOR PREVIOUS TRANSFORMATION) x u + ?

=

As in the previous run, the utiLity program GEN is used to generate a quadratic

near-identity transformation with generaL coefficients:

NORMAL FORMS

75

GEN(2); x

2

=a

2 + a

v

u v + a

1, [1, 1J

1, [0, 2J

+ U

u

1, [2, OJ

y =v + ? GEN(2) ;

2 y

=a

v

2 + a

2, [0, 2J

u v + v + a

2, [1, 1J

u

2, [2. OJ

ENTER TRUNCATION ORDER (HIGHEST ORDER TERMS TO BE KEPT) 2;

du

=v

2 + ((2 a

+ fxx) u

+ (2 a

2, [2, OJ

dt

- 4 a 2, [1, 1J

+ 2 fxy) v u 1, [2, OJ

2 + (2 a

+ fyy) v

- 2 a

2, [0. 2J

)/2 +

1, [1, 1J

2 gxx u

2 + (-

dv

4 a

+ 2 gxy) v u + (- 2 a

2, [2, OJ

+ gyy) v 2, [1, 1J

----------------------------------------------------------------- + .

2

dt

DO YOU WANT TO ENTER ANOTHER TRANSFORMATION (Y/N)

N; [VAX 8500 TIME

11 SEC. J

Now we use the utiLity program DECOMPOSE to isoLate the coefficients in the transformed equations.

Ci'[j,kJ is the coefficient of ujv

k

in the i

th

equation:

DECOMPOSE()$ [VAX 8500 TIME

4 SEC.J

We use the MACSYMA function SOLVE to eLiminate the uv and v

2

equations by appropriateLy choosing the 6 constants ai'[j,kJ'

terms in both The utiLity

program VARS is used to generate a List of the unknowns ai'[j,kJ'

Since there

are 4 aLgebraic equations in 6 unknowns, MACSYMA returns the generaL soLution in terms of 2 arbitrary constants %rl and %r2:

76

NORMAL FORMS

SOLVE([C[I,[I,I]],C[I,[O,2]],C[2,[I,I]],C[2,[O,2]]],VARS(2»; a

a

I, [2, 0]

fyy + 2 %rl

gxy

gyy + 2 fxy

[[a

2

2, [2, 0]

4

I, [I, 1]

gyy

a

a

2, [I, 1] [VAX 8500 TIME

=1

2

1,

[0, 2]

= %r2,

2 = %rl]]

a

2, [0, 2]

SEC.]

Next we substitute these values into the transformation by again calling NF.

We refer to the previous transformation as PREV, and the ai'[j,k] values as %

NF()$

DO YOU WANT TO ENTER NEW VARIABLE NAMES (YIN)?

N· DO YOU WANT TO ENTER NEW D.E.'S (YIN)?

N· INPUT NEAR-IDENTITY TRANSFORMATION (USE PREY FOR PREVIOUS TRANSFORMATION) x u + ? EV(PREV,%) ;

=

x

= %r2

2 (fyy + 2 %rl) u v (gyy + 2 fxy) u + ----------------- + ---------------- + 2 4

2 v

U

y =v + ? EV(PREV,%): 2 gyyuv gxyu + ------- + v + ------

2 y

= %rl

v

2

2

ENTER TRUNCATION ORDER (HIGHEST ORDER TERMS TO BE KEPT)

2: 2 (gxy + fxx) u = v + -------------- + dt 2

du

2 dv dt

gxx u ------ + 2

Note that the transformed equations do not depend upon the arbitrary constants

%rl, %r2.

Having accomplished the normal form computation up to quadratic

terms, we continue the process in order to include cubic terms:

NORMAL FORMS

77

DO YOU WANT TO ENTER ANOTHER TRANSFORMATION (YIN)

Y; INPUT NEAR-IDENTITY TRANSFORMATION (USE PREY FOR PREVIOUS TRANSFORMATION) x u + ? PREV+GEN(3) ;

=

3 x = a

2 + a

v

[0. 3J

1,

2 + %r2 v

u v

2 + a

1. [1. 2J

u

(fyy + 2 %r1) u v

y

3

+ ----------------- + a u 2 1. [3. OJ

y =v + ? PREV+GEN(3) ;

3

=a

v

2. [0. 3J

2 + a

v

1. [2. 1J

u v

2 (gyy + 2 fxy) u + ---------------- + 4

2 + %r1 v

2. [1. 2J

2 + a

u

gyyuv v +. ------- + v

2. [2. 1J

2

2 gxy \.i

3

+ a

u

+

2. [3. OJ ENTER TRUNCATION ORDER (HIGHEST ORDER TERMS TO BE KEPT) 3; 2 du (gxy + fxx) u + 6 gxx %r 1 + gyy gxy - 12 a v + -------------- dt 2 2. [3. OJ

«3

-- =

2

:3 gxx fyy

3 - 2 fxxx) u

+ «6 %r1 - 3 fyy) gxy - 9 fxy gyy - 12 a

- 6

fxx %r1

2. [2. 1J 2

2 - 6 fxy

- 3 fxx fyy + 12 gxx %r2 - 6 fxxy) v u

+ 36 a 1, [3. OJ

- 6 fyy fxy + 24 a

+ (- 6 fyy gyy - 24 txy %r1 - 12 a

1. [2. 1J

2. [1. 2J

2 - 12 fxx %r2 - 6 fxyy) v

u + (- 12 fyy %r1 - 12 a

- 12 %r2 fxy

2. [0. 3J 3 - 2 fyyY) v )/12 +

+ 12 a

1. [1. 2J

U

78

NORMAL FORMS 2

dv

gxx u

3

((6 fxx gxy - 6 gxx fxy - 2 gxxx) dt

u

2

+ ((- 12 gyy - 6 fxy) gxy + 36 a

+ 3 fxx gyy - 6 gxx fyy - 6 gxxy) v

2. [3. OJ 2

2 + ((- 24 %r1 - 6 fyy) gxy - 6 gyy

u

+ 24 a

-

12 gxx %r2 - 6 gxyy)

2, [2. 1J 2 v

3 u + (- 12 %r2 gxy - 12 %r1 gyy + 12 a

- 2 gyyy) v )/12 + 2. [I, 2J

00 YOU WANT TO ENTER ANOTHER TRANSFORMATION (YIN)

N;

[VAX 8500 TIME = 38 SEC.J DECOMPOSE ( ) $

[VAX 8500 TIME = 17 SEC. J Once again we use DECOMPOSE and SOLVE to obtain the ai'[j.hJ's.

The general

solution again involves two arbitrary constants. %r3 and %r4:

SOLVE([C[l,[2.1JJ,C[1.[1.2JJ,C[l,[O.3JJ.C[2.[2.1JJ.C[2.[1.2JJ.C[2.[O,3JJJ. VARS(3» ; 2

(gyy

[[a

+ 3 fxy gyy + gxyy + (2 fyy + 2 %r1) gxy - 2 %r2 gxx

1. [3, OJ 2 + fxx fyy + 2 fxy

+ 2 fxxy + 2 %r1 fxx)/12,

(4 gxy - fxx) gyy + 2 fxy gxy + 2 gxxy + 2 fyy gxx a 2, [3, OJ

12

=

a

(gyyy + (3 fyy + 6 %r1) gyy + 6 %r2 gxy + 3 fxy fyy + 3 fxyy

I, [2, 1J + 12 %r1 fxy + 6 %r2 fxx)/12, a

2. [2. 1J 2 gyy + gxyy + (fyy + 4 %r1) gxy + 2 %r2 gxx

4

NORMAL FORMS

79

fyyy + 6 %r1 fyy + 6 %r2 fxy + 6 %r3

=------------------------------------

a

1. [1. 2]

6

gyyy + 6 %r1 gyy + 6 %r2 gxy

= ----------------------------

a

2. [1. 2]

[VAX 8500 TIME

=4

= %r4,

a

= %r3]]

a

1, [0, 3]

6

2. [0. 3]

SEC.]

We terminate the calculation by substituting these vaLues into the transformed

differentiaL equations. which are stored in the variabLe NEWDES by NF:

EXPAND(EV(NEWDES,%)) ; 3 du [-dt

=v 3

2

fxxx u

3

---------- +

---------- + ------- -

12

= - ---------2

[VAX 8500 TIME

gxxy u

6

2 gxy u

6

3 fyy gxx u

---------12

2 fxx u 2

3 gxxx u

3

fxy gxy u

+ ------- + ------ + 6 2

fxx gxy u

3

fxx gyy u

3

- ----------

dt

-

+ ---------12

%r1 gxx u

dv

3

gxy gyy u

2

3 fxy gxx u

gxx u

+ ------- + ---------- + ------] 6 2 2 2 SEC.]

Note that this time the normal form does depend upon the arbitrary constant %r1.

In the case that the coeffcient gxx is not zero. %r1 can be

selected so as to eliminate the u 3 term in the u' equation. Let us apply the foregoing results to eqs.(28) which represent van der Pol's equation at infinity.

In this case

=: 2, F Fxx= 0, Fxy = -E., Fyy= 0 , Fxxx = 6E., F xyy XXY

O. Fyyy

O.

(34) G xx

O. Gxy = 0, Gyy = -4E.. Gxxx= O. G

XXY

=:

4E.. Gxyy = 4, Gyyy = O.

80

NORMAL FORMS

When these values are substituted into the results of our computer algebra work. we find:

(35.1)

u

v +

35

(35.2)

v

o+

0(4)

c u

3

+ 0(4)

Determinacy

Now the question is. have we taken enough terms of the series to correctly describe the behavior in the neightborhood of the equilibrium point?

A truncated system is said to be determined if the inclusion of any higher order terms cannot effect the topological nature of the local behavior about t~e

singularity.

For systems of the form (30). Takens [45] has shown that if

the coefficient a

2 does not vanish. then the flow of the system (30) is topologically equivalent to the flow of the simplified system

(36.1)

u'

(36.2)

v

.

=v

= a2

u

2

This result is inapplicable to eqs.(35). however. since

~

=0

there.

Takens' determinacy result (36) was extended by Rand & Keith [38] who used MACSYMA to perform a series of "blow-up" transformations necessary to draw the following conclusions: Let a

n

be the first nonzero coefficient in eq.(3Q.2).

The determinacy

results fall into two cases, depending upon whether n is even or odd:

81

NORMAL FORMS

n even: i

If all the coefficients b

i

in eq.(30.1) are zero for

< (n+2)/2.

(37)

then in some neighborhood-of the origin. the flow given by eqs.(30) is topologically equivalent to the flow given by the simplified system:

=v

(38.1)

u'

(38.2)

v' = a

n

un

This result is a natural extension of Takens' result (36).

The case where n is

odd is somewhat different. however: n odd: i

If all the coefficients b

i

in eq.(30.1) are zero for

< (n+l)/2.

(39)

and if. in addition.

(40)

4 a

n

+ n+l b

2

2

n+l

>0

"""'2 then in some neighborhood of the origin. the flow given by eqs.(30) is topologically equivalent to the flow given by the simplified system:

n+l (41. 1)

u

t

=V

+ b + u n 1

"""'2 (41.2)

v

= a

n

u

n

"""'2

82

NORMAL FORMS

In order to apply these results 'to eqs.(28), we must extend the computer algebra treatment beyond eqs.(35) to include terms of higher order, until the first a

n

coefficient is nonzero.

We omit the MACSYMA run since the

procedure is to use the functions NF. DECDMPOSE, and SOLVE just as in the previous run.

The results are as follows:

If in eqs.(2S) we perform the near-identity transformation

(42.1)

2 3 2 4+156 3 X=U-2"6U +-6-- u

(42.2)

y = v - 26UV +

+

326 3 U

246+1856 30

3 u

5

+

22724 3 3 6 u - (56+863 ) Uv

+ (1+46 ) U V -

2 4 16+2076 +1926 4 12 u v

we obtain the normal form:

(43.1)

(43.2)

v'

- 2

6

2

U

5

+ 0(6)

Eqs.(43) are of the form (30) with a

5

= -26

2

and b

3

= 56/3.

For these

values, n = 5 is odd and eq.(40) is satisfied:

(44)

and so we may conclude that in the neighborhood of the origin, the flow of (43) is topologically equivalent to the flow of the simplified system:

83

NORMAL FORMS

(45.1)

(45.2)

v

=-

2 5 2 1'0 U

Eqs.(45) may be simplified by setting q = u with dT = u

2

3

and reparameterizing time

dT (recall T was defined in connection with eqs.(26».

This

gives:

(46.1)

q

(46.2)

v' -

=5tq+3v

2

21'0

q

where primes represent differentiation with respect to T.

This linear system

has eigenvalues 2t and 31'0. and thus is an unstable node for 1'0 node for 1'0

< 0).

>0

(and a stable

In view of the preceding results on determinacy, we may

conclude that eqs.(28) are also unstable for

t

>0

(and stable for

t

< 0).

See

Fig.9 where this behavior is confirmed by numerical integration of eqs.(28). Note that it is sufficient to show that eqs.(28) are unstable for 1'0

>0

since

these equations are unchanged when 1'0, t and x are respectively replaced by

-t,

-t and -x, and thus an unstable equilibrium point becomes stable when 1'0 changes sign.

NORMAL FORMS

0.1

y

-1

1

-0.1 Fig.9. Numerical solution of eqs.(28) for

E

0.1.

x

NORMAL FORMS

85

Exercises

1. We have pointed out that the method of normal neighborhood of the equilibrium point.

form~

is valid locally in the

Thus it is not directly

applicabl~

to

finding the limit cycle in van der Pol's equation,

w"

(PI)

+ w+

EO

(w2 -1) w'

since we have seen that Lindstedt's the limit cycle for small

to be

EO

=0

metho~

gave the approximate amplitude of

~.

Nevertheless, normal forms can be used to investigate eq.(PI) by using the singular transformation

x w =-

(P2)

-.Ii.

which gives (P3)

x"

+ x -

EO

x' + x2 x' = 0

which may be written in the form

(P4.I)

x'

(P4.2)

y'

=y - x +

The system (P4) exhibits a

~opf

Y - x2

EO

y

bifurcation at the origin when

EO

= O.

Show that the linear part of eqs.{P4) may be put in the canonical form

(P5.1)

EO u' = - 0 v + ~u

(P5.2)

v' =

Ou+

2 EO 3 EO '2 v - u· v + 20 u

86

NORMAL FORMS

by the transformation (see Chapter I, eq.(29))

(P6.1)

x

=u

(P6.2)

where !J

= J1 _ :2 Then show that by adding to eqs.(P5) the equation

(P7)

o

a normal forms transformation gives the polar form:

(P8.1)

(P8.2)

S' = !J + •••

Note that eq.(P8.1) yields the approximate amplitude of the limit cycle for small c to be 2 ~, which agrees with Lindstedt's method in view of (P2).

2. Show that the following system of two coupled van der Pol oscillators can be treated by the method of normal forms:

(P9.1)

(P9.2)

87

NORMAL FORMS

Begin, as in the previous problem, by setting

(PlO)

Then set Yi

= xi'

to obtain a first order system.

Following eqs.{P6). set

(PU)

Now use the method of normal forms to show that the near-identity transformation

=fl

A~

(P12.l)

u

(P12.2)

vI

= 17 1 - 4'

(P12.3)

u2

=f 2

+

(P12.4)

v2

= 172

- 4'

(P12.5)

E.

= E.

l

+

4' E. f l A~

E.

D~

4'

E.

D~

E.

+

B~

4'

E.

B

~

1')1 -4E.172 +

C~ f 2 + 4' E. f 1 + ••• C~ 1')2 - 4' E. 17 1 + •••

gives. in polar coordinates defined by f the approximate system:

f2 +

i

r

i

cos 6 , Tl i i

=r i

sin 6 i ,

NORMAL FORMS

88

(P13.1)

(P13.2)

(P13.3)

8 • 1

(P13.4)

s . 2

AE. BE. 1 - -- - -2 2

r2

~

r1

cos(8 -8 ) + ••• 2 1

De Cc r 1 1 - -2 - -2 -r cos(S 1-8 2 ) + ••• 2

2 In deriving eqs.(P13), work to order 3 but neglect terms of O(E.).

Eqs.(P13)

were originally derived by the two variable expansion method [37] and have been studied in

[9~,[lO].

CHAP1ER 4

THE TWO VARIABLE EXPANSION METHOD

Introduction

The two variable expansion method [20J, also known as the method of multiple scales [32J, is an extension of Lindstedt's method.

While Lindstedt's

method is useful for approximating periodic solutions of o.d.e. 's, it offers no information about the stability of the solutions.

The two variable method is

used to construct an approximate general solution to a system of o.d.e. 's.

It

can be used to obtain the behavior of the system in the neighborhood of a limit cycle or an equilibrium point. In order to fix our ideas, let us again think in terms of van der Pol's equation as an example:

2

(I)

c (1 - x )

dx

dt

The method is based on the idea that the approach to a limit cycle occurs on a slower time scale than the time scale of the limit cycle itself. For example, if the motion around the limit cycle has frequency of order unity, then the approach to the limit cycle may occur on a time scale of ct.

In

particular, this is the case for Hopf bifurcations and perturbations off of Hamiltonian systems.

Algebraically this means that those terms in the solution

which represent the approach to steady state involve an explicit dependence on

90

TWO VARIABLE EXPANSION METHOD

time t in the form ct.

The method utilizes this dependence by assuming that

the solution is explicitly a function of two time variables, which we shall call f

and~.

Here f will represent ordinary time t and

~

will represent slow

time ct:

(2)

t.

The method assumes that the solution x explicitly depends upon both time variables:

(3)

Substitution of (3) into (1) requires the use of the chain rule:

(4)

dx dt

ax

= af

df c3x ~ dt + aTl dt

c3x

= BE

c3x

+ c

aTl

(5)

In addition to the change of independent variables given by (4) and (5). we expand x in a power series in c:

(6)

We note at this point that we will use this method only up to terms of order c.

Although the method can in principle be extended to arbitrary order

in c, we have found that its use beyond the lowest order terms in c is awkward. If one wanted to apply the method to higher order terms. then additional dependence on time could be built into the method by either taking new 2 independent variables such as c t. c 3 t ....• or by stretching f or~. e.g .•

TWO VARIABLE EXPANSION METHOD

with f = (1 + w e + w e 2 + ••• ) t. 1

91

See [32],[20].

2

Substituting (4)-(6) into van der Pol's eq.(I) and collecting terms, we obtain: (7)

(8)

Eq.(7) is a partial differential equation. involves two arbitrary functions of

Its general solution

T):

(9)

Substituting (9) into (8) and simplifying the trig terms gives:

(10)

+

A2 4" (A

2. B2 2 - 3B ) Sln 3f + 4" (B - 3A ) cos 3f

where primes represent differentiation with respect to

T).

Elimination of secular terms in (10) is accomplished by requiring the coefficients of cos f and sin f to vanish. differential equations on A(T)

and B(T)

This gives a pair of coupled

called the slow flow equations:

TWO VARIABLE EXPANSION METHOD

92

A'

(11.1)

(11.2)

Eqs.(ll) are most easily treated by transforming to polar coordinates,

(12)

A = R cos

a.

B

= R sin a

whereupon we obtain:

(13.1)

R'

(13.2)

8'

3 R

R

= 2" - 8 o

Now eq.(l3.l) is a separable first order o.d.e.

2 R(O) e~/2

(14)

J (e~ -

1) R(O)2 + 4

This eq. has the property that as

n

~

It is easily solved:

_00,

R

~

0 if R(O)

< 2,

R

~ ~oo,

~

2, the limit cycle amplitude.

As

so that if we run time backwards. all points inside

the limit cycle approach the equilibrium at the origin. the denominator has a zero at n

= log[l

2 - 4IR(O) ]

< O.

If R(O)

> 2,

however,

Thus if we run time

backwards, motions starting outside the limit cycle are predicted to escape to infinity in finite time. in agreement with numerical investigations. In terms of Rand 8, eq.(9) for X

o becomes

(15)

TWO VARIABLE EXPANSION METHOD

93

Thus the two variable expansion method to O(c) yields the result that the limit cycle in van der Pol's equation is stable.

Computer Algebra

We shall present a MACSYMA program which wi 11 apply the two variable expansion method to a nonautonomous (i.e. forced) system of n coupled oscillators.

We begin by giving a sample run on the van der Pol equation.

Then we give a listing of the program and follow it with a discussion of the more general problem of n coupled oscillators:

TWOVARO; 00 YOU WANT TO ENTER NEW DATA (YIN) y; NUMBER OF D.E.

's

1;

THE 1 D.E. 'S WILL BE IN THE FORM: X[I]" + W[I]A2 X[I] = E F[I](X[l], ... ,X[ 1 ],T) ENTER SYMBOL FOR X[ 1 ] x;

ENTER W[ 1 ] 1;

ENTER F[ 1 ] (1-x A2)*diff(x,t); THE D. E. ' S ARE ENTERED AS: 2

dx

x"+x=e(1-x) dt THE METHOD ASSUMES A SOLUTION IN THE FORM: X[I] = XO[I] + E Xl[I] WHERE XO[I] = A[I](ETA) COS WEI] XI + B[I](ETA) SIN WEI] XI WHERE XI = T AND ETA = E T REMOVAL OF SECULAR TERMS IN THE Xl[I] EQS. GIVES: 2 a d

2 (---- (a

deta

1

»+

3 a

b

1

1

1

----- + -- - a

4

4

3 b

deta

» 1

4

=0

2 b 1 1 - ----- + b

a

1

d - 2 (---- (b

1

4

=0 1

TWO VARIABLE EXPANSION METHOD

94

00 YOU WANT TO TRANSFORM TO POLAR (X)()RDINATES (YIN) y;

3 r

d

r

1

[[---- (r ) deta 1

2

1

8

d

deta

(theta) = 0]] 1

00 YOU WANT TO SEARCH FOR RESONANT PARAMETER VALUES (YIN) n;

[VAX 8500 TIME = 16 SEC.]

Here is the program listing:

TWOVAR():=BLOCK( CHOICE:READ("oo YOU WANT TO ENTER NEW DATA (YIN)"), IF CHOICE = N THEN GO(JUMP1),

1* INPUT D.E. 's *1 N:READ("NUMBER OF D.E. 'S"), PRINT("THE",N,"D.E.'S WILL BE IN THE FORM:"), PRINT("X[I]" + W[IY2 XCI] = E F[I](X[l], ... ,X[",N,"],T)"), FOR I: 1 THRU N 00 XCI] :READ("ENTER SYMBOL FOR X[", I, "]"), FOR 1:1 THRU N 00 DEPENDS(X[I],T), FOR I: 1 THRU N 00 W[I]:READ("ENTER W[",I,"]"), FOR I: 1 THRU N 00 F[I]:READ("ENTER F[",r."]"),

JUMP1,

1* UPDATE EQS FOR SUBSTITUTION OF RESONANT VALUES ON 2ND TIME THRU*I FOR 1:1 THRU N oo( W[I]:EV(W[I]),

TWO VARIABLE EXPANSION METHOD F[I]:EV(F[I])). /* ECHO BACK THE D.E. 'S */ PRINT("THE D.E. 'S ARE ENTERED AS:"), FOR I: 1 THRU N DO PRINT(X[I] ... ·· +".W[I]A2*X[I],"=".E*F[I]), PRINT("THE METHOD ASSUMES A SOLUTION IN THE FORM:"), PRINT("X[I] = XO[I] + E Xl[I]"), PRINT("WHERE XO[I] = A[I](ETA) CDS Wei] XI + B[I](ETA) SIN Wei] XI"). PRINT("WHERE XI = T AND ETA = E T"). /* DON'T USE A OR B AS PARAMETERS IN THE GIVEN D.E. 'S */ DEPENDS([A,B] .ETA) , FOR I: 1 THRU N DO XO[I]:A[I]*CDS(W[I]*XI)+B[I]*SIN(W[I]*XI), FOR I: 1 THRU N DO G[I]:EV(F[I].T=XI). FOR I: 1 THRU N DO( FOR J: 1 THRU N DO G[I]:EV(G[I].X[J]::XO[J])). FOR I: 1 THRU N DO( G[I]:G[I]-2*DIFF(XO[I].XI.1.ETA.1). G[I]:EV(G[I].DIFF). G[I]:EXPAND(TRIGREDUCE(EXPAND(G[I])))), /* CDLLECf SECULAR TERMS */ FOR I: 1 THRU N DO( S[I]:CDEFF(G[I].SIN(W[I]*XI)). C[I]:CDEFF(G[I].CDS(W[I]*XI))). /* DISPLAY SECULAR TERMS */ PRINT("REMOVAL OF SECULAR TERMS IN THE Xl[I] EQS. GIVES:"). FOR I: 1 THRU N DO(

TWO VARIABLE EXPANSION

96

METHOD

PRINT(S[I]."= 0"). PRINT(C[I],"= 0"», ABEQS:APPEND(MAKELIST(S[I].I.I.N),MAKELIST(C[I],I.I,N». CHOICE2:READ("OO YOU WANT TO TRANSFORM TO POLAR COORDINATES (YIN)"). IF CHOICE2=N THEN GO(JUMP2), 1* TRANSFORM TO POLAR COORDINATES *1

DEPENDS([R. THETA] ,ETA). TRANS:MAKELIST([A[I]=R[I]*CDS(THETA[I]),B[I]=R[I]*SIN(THETA[I])].I,I.N), INTEQS: EV(ABEQS, TRANS. DIFF) • RTHEQS:SOLVE(INTEQS,APPEND(MAKELIST(DIFF(R[I].ETA);I,I,N). MAKELIST(DIFF(THETA[I],ETA),I,I.N»). RTHEQS:TRIGSIMP(RTHEQS). RTHEQS: EXPAND (TRIGREDUCE (RTHEQS) ) . PRINT (RTHEQS) ,

JUMP2. CHOICE3:READ("OO YOU WANT TO SEARCH FOR RESONANT PARAMETER VALUES (YIN)"). IF CHOICE3=N THEN GO(END), 1* OBTAIN FREQUENCIES WHICH APPEAR ON RHS'S OF D.E. 's *1 1* DEFINE PATTERN MATaIING RULES TO ISOLATE FREQS *1

MATCHDECLARE( [DUMMYI. DUMMY2]. TRUE) , DEFRULE(mSARG,DUMMYI*COS(DUMMY2) ,DUMMY2). DEFRULE(SINARG,DUMMYl*SIN(DUMMY2).DUMMY2). FOR I: I THRU N OO( 1* CHANGE SUM TO A LIST *1

TEMPI:ARGS(G[I]), 1* REMOVE mNSTANT TERMS *1

TEMP2:MAP(TRIGIDENTIFY,TEMPl),

TWO VARIABLE EXPANSION METHOD f* ISOLATE ARGUMENTS OF TRIG TERMS *f

TEMP3:APPLY1(TEMP2.00sARG.SINARG). TEMP4:EV(TEMP3.XI=1), f* REMOVE DUPLICATE ARGUMENTS *f

FREQ[I]:SETIFY(TEMP4», f* DISPLAY FREQUENCIES *f

FOR I: 1 TIIRU N DO( PRINT(X[I], "EQ'S RESONANT FREQ =", W[I]), PRINT("FREQS ON RHS =",FREQ[I]»,

JUMP3, PAR:READ("WHICH PARAMETER TO SEARCH FOR ?"). f* COMPlITE RESONANT VALUES OF DESIRED PARAMETER *f

RESVALS: [] , FOR I: 1 TIIRU N DO( FOR J:1 THRU LENGTH(FREQ[I]) DO( RES:SOLVE(PART(FREQ[I],J)=W[I],PAR), IF (RES#[] AND RES#ALL) THEN RESVALS:APPEND(RESVALS,RES), RES:SOLVE(PART(FREQ[I],J)=-W[I].PAR), IF (RES#[] AND RES#ALL) THEN RESVALS:APPEND(RESVALS,RES»), f* ELIMINATE DUPLICATE PARAMETER VALUES *f

RESVALUES:SETIFY(RESVALS) , f* DISPLAY RESONANT PARAMETER VALUES *f

PRINT(RESVALUES), CHOICE4:READ("DO YOU WANT TO SEARCH FOR ANOTHER PARAMETER (YIN) ?"), IF CHOICE4=Y THEN GO(JUMP3), END," ")$

TWO VARIABLE EXPANSION METHOD

98

1* AUXILIARY FUNCfIONS *1 TRIGIDENTIFY(EXP) :=IF FREEOF(SIN ,EXP) AND FREEOF(<XlS.EXP) THEN 0 ELSE EXP$

SETIFY(LIST): =( SET: [LIST[ 1]], FOR I: 2 THRU LENGTH(LIST) OO( IF NOT MEMBER(LIST[I] ,SET) THEN SET:CONS(LIST[I].SET». SET) $

Systems of Coupled Oscillators

Our main interest in the two variable method will be in applications to systems of n oscillators:

(16)

After replacing time t by the two independent variables f and

(17)

f

= t.

~

=c

~.

t

and expanding each of the dependent variables xi in a power series in c,

we obtain the following differential equation on xiI:

(19)

TWO VARIABLE EXPANSION METHOD

99

where (cf. eq.(9»

(20)

A.(D) cos w.f sin w1.f 1 1 + B.(D) 1

The program removes secular terms from (19) and offers the user the option of transforming to polar coordinates as in the preceding example.

In

addition. the program offers the user the option of searching for resonant parameter values. i.e .• parameter values which cause an innocuous term on the right hand side of (19) to become a resonant term.

Since we are only working

to O(c) throughout this chapter. the utility of this approach will be most significant in problems involving several oscillators and/or several parameters.

This is in contrast to situations in which one is concerned with

obtaining many terms of a series solution. in which case computer size and speed limitations usually prohibit problems which involve more than one or two parameters. As an example. we will consider the following system of two oscillators:

(21. 1)

(21.2)

d

2

~ + (1 + c Ax) x + c v x 2 dt

3

c k y

2 d ~ + (1 + c A ) y + c ~ y cos wt

dt 2

y

o

This system may be described as a Duffing oscillator (x) driven by the output from a Mathieu equation (y). which in turn drives the x equation.

The cos wt term drives the y equation The parameters are:

100

TWO VARIABLE EXPANSION METHOD A x

A = detuning coefficients y

v

= nonlinearity coefficient

~

=

forcing coefficient

w

forcing frequency

k

= coupling coefficient

Here is the MACSYMA run annotated with comments in italics:

TWOVARO; DO YOU WANT TO ENTER NEW DATA (YIN) y;

NUMBER OF D.E. 'S 2;

THE 2 D.E. 'S WILL BE IN THE FORM: X[I]" + W[I]A2 XCI] = E F[I](X[l], ... ,X[ 2 ],T) ENTER SYMBOL FOR X[ 1 ] x;

ENTER SYMBOL FOR X[ 2 ] y;

ENTER W[ 1 ] l'

ENTER W[ 2 ] 1;

ENTER F[ 1 ] -delx*x-nu*x A 3+k*y;

ENTER F[ 2 ] -dely*y-mu*y*cos(w*t);

THE D. E. 'S ARE ENTERED AS: 3

x

+ x

=

y

+ Y

=e

e (k y - nu x

- delx x)

(- mu cos(t w) y - dely y)

THE METHOD ASSUMES A SOLUTION IN THE FORM: XCI] = XO[I] + E X1[I] WHERE XO[I] = A[I](ETA) COS WeI] XI + B[I](ETA) SIN WeI] XI WHERE XI = T AND ETA = E T REMOVAL OF SECULAR TERMS IN THE X1[I] EQS. GIVES: 2

3

nu

3 b

1

3 a

nu

b

1

d

1

- ---------- + b

4

4

k - b

2

delx + 2 (---- (a » deta 1 1

=0

TWO VARIABLE EXPANSION METHOD 2 3 a

3

b

1

101

nu

3 a

1

nu

1

d

k - a

+ a

4

4

2

delx - 2 (---- (b )) 1 deta 1

o

d

2 (---- (a )) - b dely deta 2 2 d

- a

dely - 2 (---- (b )) 2 deta 2

0

=0

DO YOU WANT TO TRANSFORM TO POLAR COORDINATES (YIN)

n;

We wait to transform to polars until after we have searched for resonant parameter values:

DO YOU WANT TO SEARCH FOR RESONANT PARAMETER VALUES (YIN)

y; x EQ'S RESONANT FREQ = 1 FREQS ON RHS = [1, 3] y EQ' S RESONANT FREQ = 1 FREQS ON RHS = [I, w - I, w + 1]

In order for a term on the right hand side (RHS) of eq.(19) to be resonant, its frequency must equal Wi'

The program finds all values of the

selected parameter which produce a match between a resonant frequency Wi and an

element in the associated list of frequencies appearing on the RHS:

WHICH PARAMETER TO SEARCH FOR ?

w;

[w

= -2,

w = 0,

W

= 2]

DO YOU WANT TO SEARCH FOR ANOTHER PARAMETER (YIN) ? n;

[VAX 8500 TIME

= 14

SEC.]

Having obtained a list of resonant driver frequencies w, we may investigate the effect of choosing one of them:

w:2$

TWO VARIABLE EXPANSION METHOD

102

We selected the resonant value 2 for w and assigned w this value.

Now

we again call the program TWOVAR which automatically inserts this new value of w into the previous equations:

TWOVAR() ; 00 YOU WANT TO ENTER NEW DATA (Y/N) n; THE D. E .• S ARE ENTERED AS: 3

x

+ x

=e

(k y - nu x

y

+ Y

=e

(- mu cos(2 t) y - dely y)

- delx x)

THE METHOD ASSUMES A SOLUTION IN THE FORM:

X[I] = XO[I] + E X1[I] WHERE XO[I] = A[I](ETA) COS WEI] XI + B[IJ(ETA) SIN WEI] XI WHERE XI = T AND ETA = E T REMOVAL OF SECULAR TERMS IN THE X1[I] EQS. GIVES:

3

2

3bnu 1

3abnu 1 1

d

- ------- - ---------- + b

4

4 2

3 a

b

k - b 2

delx + 2 (---- (a » deta 1 1

=0

3

nu

3 a

nu 1 1 1 d - ---------- - ------- + a k - a delx - 2 (---- (b » 4 4 2 1 deta 1

=0

mu

b

2 ----- - b

2

d

dely + 2 (---- (a » 2 deta 2

=0

mu

a 2

d

- ----- - a dely - 2 (---- (b » 2 2 deta 2

=0

00 YOU WANT TO TRANSFORM TO POLAR COORDINATES (Y/N) y;

r sin(theta - theta) k d 2 2 1 [[---- (r ) - - ------------------------deta 1 2

r

2

d

deta

(r ) 2

sin(2 theta ) mu 2 4

TWO VARIABLE EXPANSION METHOD

103

2

3 r

nu

r

1

d

(theta) deta 1

cos(theta 2

2

- theta) k 1

delx

- ------- + ------------------------- 8 2 r

2

1

cos(2 theta ) mu d

(theta) deta

dely

2

=4

2

- ----]] 2

DO YOU WANT TO SEARCH FOR RESONANT PARAMETER VALUES (Y/N) n;

[VAX 8500 TIME

= 91

SEC.]

The program has obtained the slow flow equations corresponding to the resonant value w

= 2.

As always in this method there is still the problem of

dealing with the slow flow equations.

Since the y equation drives the x

equation (but not vice versa), the slow flow equations on A and B are 2 2 uncoupled from the rest:

(22.1)

(22.2)

B'

2

Differentiating (22.1) and substituting in (22.2) produces the equation:

(23)

2

Thus the system (22) has bounded solutions if and only if A 2 - ~ y

if and only if

(24)

4

> 0, i.e.,

TWO VARIABLE EXPANSION METHOD

104

This yields the well known result [44J that for small forcing amplitudes, Mathieu's equation (21.2) possesses a resonance instability when the forcing frequency is approximately twice the natural frequency.

In fact. an infinite

number of such regions of instability exist when w = 2/n. for n = 1,2.3, ... , n but the perturbation method must be extended to O(c ) in order to pick up the n

th

such region [35J. When eq.(24) holds. or when the forcing frequency w is not resonant.

the general solution (20) to Mathieu's equation (21.2) is composed of terms which are the product of a slowly varying periodic function (A periodic function of frequency unity (cos t or sin t).

or B ) and a 2

2

This results. in

general. in an almost periodic function. We now consider the full slow flow system of 4 differential equations in the resonant case (w

= 2).

In polar form these were found to be:

(25.1)

8 . 1

(25.2)

(25.3)

8 . 2

(25.4)

These equations are too complicated to hope for an exact general solution.

Instead we will look for equilibria.

Note that an equilibrium of

the slow flow system (25) corresponds to a periodic solution of the original system (21) (cf. eq.(20». In order for eq.(25.3) to have

an

equilibrium, either r

2

= 0 (in which

case the y oscillator is turned off so that the problem becomes the free

TWO VARIABLE EXPANSION METHOD

105

undamped Duffing equation, which is integrable). or sin 282 case, cos 282

~ ~

~

O.

1. and for equilibrium. eq.(25.4) requires A y

In the latter' ~

JL

2" '

a

condition corresponding to the transition between stability and instability in the Mathieu equation on y (cf. eq.(24)).

Requiring this condition to hold, we

set the right hand sides of eqs.(25.l) and (25.2) to zero for equilibrium. Eq.(25.1) requires either r

2

~

In the latter case, cos(82-81)

0 (the integrable case again) or sin(8 -8 ) 2 1

=~

O.

1. and eq.(25.2) requires

(26)

This equation relates the amplitude of the derived periodic motion r to the amplitude of the driver r A

y

2

1

and the other parameters in the case that

+ !:!:. - 2

Exercises

1.

Find the slow flow associated with the dynamics of two coupled van der Pol

oscillators [37J,[10J:

2 2 dx d x - - + X - e (1 - x ) 2 dt dt

e k

2 2 d -L+ y - e (1 - Y ) dy = e 2 dt dt

l1

~1

x + e k

12

y + e c

l1

dy dx + e c 12 dt dt

dy x + e k 22 y + e c 21 dx + e c 22 dt dt

106

2.

TWO VARIABLE EXPANSION METHOD

Find the slow flow associated with the dynamics of n coupled van der Pol

osc i llators:

2 d x.

__ 1

+

dt 2

Xi -

Eo

= 1.2 ..... n

where i

and where kO

consider the cases n

= 2.3.4.

= kn = O.

The procedure here is to separatel

etc .• until a general pattern emerges regarding

the structure of the slow flow equations.

3. Apply the two variable expansion method to the equation

::~ +

(PI)

in which

[c} +

Eo

A + c cos t] x +

Eo C

:

+ c a x

3

=0

w = natural frequency A c

detuning coefficient

= damping

coefficient

a = nonlinearity coefficient

Show that w flow.

=~

is a resonant parameter value.

Obtain the associated slow

Find a relation between A and c such that the trivial solution of the

slow flow (which corresponds to the rest solution x

=0 of

(PI»

is unstable.

The nontrivial equilibria of the slow flow correspond to 2:1 subharmonic periodic solutions of (PI).

Find an expression for the amplitude r of such

motions as a function of A. c and a.

For given a. find a relation between A

and c such that 2:1 subharmonics exist.

CHAPTER 5

AVERAGING

Introduction

Like normal form transformations, averaging uses a near identity coordinate transformation to simplify a given system of ordinary differential equations.

In contrast to our treatment of normal forms, where we applied the

method to strictly autonomous systems. we shall apply averaging to nonautonomous systems.

The coordinate transformations will be chosen so as to

transform the nonautonomous system into an autonomous one called the averaged system. Consider the system

(I)

x

e f{x. t.e) •

where f is T-periodic in t. with respect to averaging.

Such a system will be said to be in general form Since e

«

1. x usually evolves on a much slower

time scale than does f. which we can consider as T-periodic forcing.

Therefore

it seems natural that the main influence for the time evolution of x comes from the mean of f over one period T.

This is· the basic idea of averaging which

leads to the following algorithm [14].[41],[16]:

(2)

Set

AVERAGING

108

and insert into (1). Then we find

(3)

awl ] 2 e [ gl(y,t) - --+ O(e )

(4)

at

where D is the Jacobian matrix operator, and where y

(5)

fey, t,O).

We split gl(y,t) into a mean component gl(y) and a time dependent component gl(y,t) with zero mean,

~

(6)

gl(y,t) = gl(y) + gl(y,t),

where

=~

(7)

T

J f(y,t,O) dt,

TO

and where gl(y,t) is given by the difference between eqs.(5) and (7).

Then we

can choose w (y,t) such that 1

(8)

where the arbitrary constant of integration in (8) is chosen such that wI has zero mean.

This gives

AVERAGING

(9)

109

y

Eq.(9) is the first order averaged system corresponding to equation (1).

There are several theorems concerning the relation between the averaged

and the general system (for details. see e.g. [41]).

Since we are mainly

interested in the computational aspects of the perturbation method we shall not state these theorems here explicitly. but we shall try to give the general idea of their content.

One can prove that a solution yet) of the averaged system

(9) follows a true solution x(t) of the general system (1) for a time of order 1/~

if the initial values Yo and

IYO-xol

= O(~).

X

o

were close to order

~.

i.e. if

Also all the qualitative local behavior of the dynamics of (9)

corresponds to the same qualitative and local behavior of periodic orbits of (1).

In particular a stable (unstable) fixed point of (9) corresponds to a

stable (unstable) limit cycle of (1). and a Hopf bifurcation giving rise to an attracting (repelling) limit cycle in (9) corresponds to a bifurcation to a stable (unstable) invariant torus in (1). and so on.

There are additional

results concerning global information on periodic orbits of (9) for which we refer the reader to [14].

Second Order Averaging

It frequently happens that the system (9) is degenerate and does not provide enough information about the solutions of (1).

In such a case the

averaging method can be extended to include order ~2 terms by replacing the transformation (2) with

(10)

Substitution of (10) into (1) gives

110

AVERAGING

where gl(y,t)

f(y.t.O) as in (5). and where

We split g2(y.t) into a mean component g2(y) and a time dependent ~

component g2(y.t) with zero mean.

~

(l3)

g2(y,t}

= g2(y}

+ g2(y·t}.

where

J[-at=. T

= -I T 0

( 14)

af (y.t.O) + D~I(y.t) ~ wI

~

and where g2(Y.t) is given by the difference between eqs.(12} and (14).

We

have chosen w (y.t) as in (8). and have used the fact that 1 f(y.t.O)

= gl{y.t) = gl{y}

zero mean.

+ gl(y·t) and that Dyw1 gl{y} and Dygl(y} wI have

Then we can choose w2 (t} such that

(15)

where the arbitrary constant of integration in (IS) is chosen such that w2 has zero mean.

(16)

This gives

AVERAGING Although we shall only work to order e

111 2

in this Chapter, the method of

averaging can be extended in a similar fashion to include terms of order em. See Exercise 1 at the end of this Chapter.

Van der Pol Transformation

Averaging is frequently used to treat weakly nonlinear forced oscillators of the form

(17)

where W is the forcing frequency.

Although this important class of

applications is not of the form (1). it may be put in general form with respect to averaging by the following procedure. The unperturbed system (e

= 0)

has the general solution

(18)

where Xl and x problem.

2

are constants given by the initial values of the unperturbed

Allowing for Xl and x

of parameters). and replacing

2

to be time dependent (the method of variation

o by

W

an

a~

yet unspecified frequency WI' we

obtain the van der Pol transformation [14]

(19)

[:]

=

cos(W t) l WI sin(w 1 t)

With this transformation, eq.(17) becomes

- sin(w l t) ] [Xx I ] - WI cos(wlt) 2

AVERAGING

112

(20) 2 2 . ] [ (w 1 - Wo ) z + c F(z,z,t,w,c) cos(w 1 t)

A natural choice for w is the forcing frequency w. 1

However, in order

to include the case of subharmonic resonance of order k, i.e. the case where w

= k wo '

we choose w 1

= w/k.

Allowing for an order c detuning off of the

subharmonic resonance, we set c

n = w2

2 2 - k W

o

With these substitutions, (20)

is in general form with respect to averaging, cf. eq.(l):

k

- c

xl

w

n k

z + F(z,~,t,w,c)

sin(wt/k)

2

(21) - c k

x2 =

w

n

-

k

z + F(z,~,t,w,c) ] cos(wt/k)

2

where z and z are given by (19).

An Example

As an example, let us take van der Pol's equation,

(22)

z

+ z

Performing the van der Pol transformation (19) with w 1 (20),

2

•

Xl

- c (1 - z ) z sin t

x

2 • - c (1 - z ) z cos t

(23)

2

1 gives, from

AVERAGING where

Z

= xl

cos t -

X

2

sin t.

113

Now we may apply the transformation (2). with

WI as in (8). which turns out to give the averaged equations (cf. (9»:

(24)

Eqs.(24) may be simplified by transforming to polar coordinates,

(25)

r sin 8

r cos 8.

which gives

o

(26)

Equilibria for eq.(26) are r

=0

and r

= 2.

The latter recovers the small e

approximation for the limit cycle in van der Pol's equation (22).

Computer Algebra

The MACSYMA program which we shall use to implement averaging takes a system of the form (1) and transforms it to the first order averaged form (9) or the second order averaged form (16). In the case of first order averaging this requires only that gl(y) be calculated from eq.(7).

Instead of computing the integral in (7). however, we

use a more efficient scheme for eliminating the periodic terms.

We convert all

sines and cosines to complex exponentials (a step performed by the MACSYMA function EXPONENTIALIZE), algebraically expand the result and then kill all terms which contain exponentials in t (a step performed using a pattern matching rule called KILLEXP).

AVERAGING

114

In the case of second order averaging we must also compute g2{y) from eq.(14).

This involves finding wI from (8).

Periodic terms are again removed

by using pattern matching instead of integration. In order to conveniently treat weakly nonlinear oscillators of the form (17). we include the van der Pol transformation in the program. Before giving a sample run and the program listing. we mention two caveats to the user: 1. 2~

The user must scale time so that averaging is performed over period

(except in the case of a weakly nonlinear oscillator (17). where this is

done automatically.) 2.

Our scheme for replacing integration by elimination of exponentials

only works for that class of problems in which all trigonometric terms occur in the numerator of the vector field f{x.t.c) in (I).

In other cases. the program

may be easily changed to use integration to compute gl{y) and g2{y) in (7) and (14) .

Here is a sample run on the van der Pol oscillator (22):

AVERAGE() ; DO YOU WANT TO ENTER A NEW PROBLEM? (Y/N)

Y; ARE YOU CXlNSIDERING A WEAKLY NONLINEAR OSCILLATOR OF THE FORM Z" + OMEGAO 2 Z = EPS F{Z.ZIlOT.T) ? (Y/N) A

Y;

ENTER OMEGAO I;

ENTER F{Z.ZDOT.T) {1-Z 2)*ZDOT; DO YOU WANT FIRST OR SECXlND ORDER AVERAGING? A

(1/2)

1;

2 y1 y2 8

8

2

3

y1

- --- + --)

[eps ([VAX 8500 TIME

3

y1

2

. eps

y2

y1

y2

y2

- ------ + --)]

(8

8

2

25 SEC.]

This result is the same as eq.(24) given in the text.

Before

converting to polar coordinates (25). we run the program again to include

AVERAGING second order effects.

115

Note that the problem need not be reentered:

AVERAGE() ; DO YOU WANT TO ENTER A NEW PROBLEM? (YIN)

N; DO YOU WANT FIRST OR SECOND ORDER AVERAGING?

(1/2)

2;

2 matrix([ eps

5 2 3 3 4 2 11 y2 11 y1 y2 3 y2 11 y1 y2 3 y1 y2 y2 (------ + ---------- - - - - - - + --------- - - - - - - - - - + --) 256

2

16

3

y1 y2

y1

+ eps (-

y1

2

- --- + --)], [eps

8

8

5 16

256 y2

- --) + eps (-

16

8

128

16

2

3

y1 8

256

4 3 2 2 11 y1 y2 11 y1 y2 3 y1 y2 (- --------- - ---------- + --------

2

3

11 y1 3 y1 - ------ +

256

128

y1

y2

y2

- ------ + --)])

8

8

2

[VAX 8500 TIME = 68 SEC.] In order to transform to polar coordinates, we use the program TRANSFORM given in Chapter 2.

The output of the program AVERAGE is stored in a

variable called RESULT:

TRANSFORM() ; Enter number of equations 2;

Enter symbol for original variable 1

n;

Enter symbol for original variable 2 Y2; Enter symbol for transformed variable

R; Enter symbol for transformed variable 2 THETA; The RHS's of the d.e. 's are functions of the original variables: Enter RHS of y1 d.e. d y1 /dt = RESULT[1,1]; 5 4 2 2 3 3 3 y1 y2 y2 11 y1 y2 3 y2 11 y1 y2 2 11 y2 d y1 /dt = eps (------ + ---------- - ----- + --------- - -------- + --) 16 8 256 128 16 256 2

y1 y2

3

y1

+ eps (-

y1 + --)

8

8

2

AVERAGING

116

Enter RHS of y2 d.e. d y2 /dt = RESULT[2,1J; 2

d y2 /dt

eps

3 2 4 2 5 3 11 y1 y2 11 y1 y2 3 y1 y1 3 y1 y2 11 y1 (- --------- - ---------- + -------- - - - - - - - + ----- - --) 128 256 16 256 8 16 2

3

y2

y1

y2

+ eps (-

y2 + --)

8

8

2

The transformation is entered next: Enter y1 as a function of the new variables y1 = R*COS(THETA) ; y1 = r cos(theta) Enter y2 as a function of the new variables y2 = R*SIN(THETA) ; y2 = r sin(theta) 3

dr [[-dt

2

eps r

3

sin (theta) + eps r

2

cos (theta) - 4 eps r

8

dtheta

2

4 2 4 2 2 2 sin (theta) + (22 eps r cos (theta) - 48 eps r )

4

- (11 eps

r

dt 224 4 2 2 2 2 sin (theta) + 11 eps r cos (theta) - 48 eps r cos (theta) + 32 eps )/256JJ [VAX 8500 TIME

=2

SEC.J

In order to simplify the algebra here we use the MACSYMA function TRIGSIMP:

TRIGSIMP(%) ; 3

dr [[-dt [VAX 8500 TIME

eps r

- 4 eps r 8

dtheta dt

2 2 2 2 4 11 eps r - 48 eps r + 32 eps ---------------------------------JJ 256

13 SEC.J

2 This result is the 0(c ) version of eq.(26). cycle r = 2. this result gives

e= -

Note that on the limit

2 ~6 . in agreement with the frequency

2 1 - ~6 given by Lindstedt's method. cf. Chapter 1.

AVERAGING

117

Here is the program listing:

1* PROCRAM TO PERFORM 1ST OR 2ND ORDER AVERAGING

ON AN N-DIMENSIONAL SYSTEM OF NONAUTONOMOUS ODE'S *1 1* AVERAGING IS PERFORMED BY CONVERTING TRIG TERMS TO

COMPLEX EXPONENTIALS, THEN KILLING EXPONENTIALS

*1

AVERAGE(}:=BLOCK( CHOICE1 :READ("DO YOU WANT TO ENTER A NEW PROBLEM? (YIN}"), IF CHOICEI = N THEN GO(JUMP1}, KILL(X} , PRINT("ARE YOU CONSIDERING A WEAKLY NONLINEAR OSCILLATOR OF THE FORM"}, CHOICE2:REAP("Z"

+ OMEGAO A 2 Z = EPS F(Z,ZDOT,T} ? (YIN}"),

IF CHOICE2 = N THEN GO(JUMP2}, 1* ENTER DATA FOR SINGLE OSCILLATOR PROBLEM *1

N:2, OMEGAO:READ("ENTER OMEGAO"}, F:READ("ENTER F(Z,ZOOT,T}"}*EPS, 1* DOES F(Z,ZDOT,T} DEPEND EXPLICITLY ON T? *1

TEST:DIFF(F, T}, IF TEST=O THEN OMEGA1:OMEGAO ELSE ( OMEGA:READ("ENTER THE FORCING FREQUENCY"}, K:READ("ENTER THE ORDER OF THE SUBHARMONIC RESONANCE"}, OMEGA1 :OMEGAIK}, 1* VAN DER POL TRANSFORMATION *1

ZSUB:[Z=COS(OMEGA1*T}*X1-SIN(OMEGA1*T}*X2, ZDOT=-OMEGAl*SIN(OMEGA1*T}*X1-OMEGAl*COS(OMEGA1*T}*X2], 1* SUBSTITUTE ZSUB INTO TRANSFORMED EQS *1 1*

SCAL~

TIME SO THAT AVERAGING OCCURS OVER PERIOD 2 PI *1

AVERAGING

118

VF:EV([-1/0MEGA1 2*CEPS*KAPOMEGAlK 2*Z

+

F)*SIN(OMEGA1*T).

-1/0MEGA1 2*(EPS*KAPOMEGAlK 2*Z

+

F)*COS(OMEGA1*T)J.

A

A

A

A

ZSUB. T=TAU/OMEGA1. INFEVAL). VF:EV(VF,TAU=T). IF OMEGA1

#

OMEGAO

THEN PRINT( "WE WRITE EPS*KAPOMEGA =". OMEGA 2-K 2*OMEGAO 2) A

A

A

ELSE VF: EV(VF .KAPOMEGA=O) . VF2:EXPAND(EXPONENTIALIZE(VF». FOR 1:1 THRU 2 DO VF2[IJ:MAPCFACfOR.VF2[IJ).

x: [Xl ,X2]. GO(JUMP1) .

JUMP2. /* ENTER DATA FOR GENERAL PROBLEM OF N ODE'S */ N:READC"ENTER NUMBER OF DIFFERENTIAL EQUATIONS"). X:MAKELIST(CONCAT(·X.I).1.1.N). PRINT("THE ODE'S ARE OF THE FORM:"). PRINT("DX/DT = EPS F(X.T)"), PRINT("WHERE X =" .X). PRINT("SCALE TIME T SUQI THAT AVERAGING OCCURS OVER 2 PI"), VF:MAKELIST(READC"ENTER RHSC", I, ")=EPS* ... ")*EPS. I, I.N). FOR 1:1 THRU N DO PRINTC"D",X[I],"/DT =",VF[I]), VF2:EXPANDCEXPONENTIALlZECVF» . FOR 1:1 THRU N DO VF2[IJ:MAP(FACfOR.VF2[IJ).

JUMP1. /* AVERAGING */ M:READC"DO YOU WANT FIRST OR SECOND ORDER AVERAGING? COEFVFEPSl:COEFF(VF2,EPS).

C1I2)").

AVERAGING

119

alEFVFEPS2: alEFF(VF2. EPS. 2) . G1BAR:DEMOIVRE(APPLY1(alEFYFEPS1,KILLEXP». IF M=1 THEN RESULT:EPS*G1BAR ELSE(

G1TILDE:alEFVFEPS1-G1BAR. W1:INTEGRATE(G1TILDE.T). REMARRAY(JAalB) . JAalB[I.J] := DIFF(G1TILDE[I].X[J]). JAOG1TILDE:GENMATRIX(JAalB.N.N). G2BAR:DEMOIVRE(APPLY1(EXPAND(JAOG1TILDE.W1)+alEFVFEPS2.KILLEXP». RESULT: EPS*GIBAR+EPS~2*G2BAR) . 1* REPLACE X BY Y *1

FOR 1:1 THRU N DO RESULT:SUBST(alNCAT('Y.I).CDNCAT('X.I).RESULT), RESULT) $

1* AUXILIARY FUNCTIONS TO KILL EXPONENTIAL TERMS *1

CDNTAINS_T(EXP) := NOT FREEOF(T. EXP)$ MATCHDECLARE(Q.alNTAINS_T)$ DEFRULE(KILLEXP.EXP(Q).O)$

Additional Examples

As our second example. we consider a modification of the van der Pol equation (22) (from [41]):

(27)

•• z +

Z

2 2 2· = c a z + c (1-z) z

Here the first order mean is zero, and we must appeal to second order averaging for a nontrivial result.

Application of our program AVERAGE as in the

AVERAGING

120 preceding example gives the RESULT:

[ [ [

2 eps

2 3 5 alpha y2

(------------ -

2 y1 y2

2 5 alpha

2 yl

3

y2

y1

y1

+ --------------- - --- + --) 8 2

8 12 [ 12 [ 3 2 2 2 [ y2 y1 y2 5 alpha y1 y2 2 y2 [ --------------- - ------ + -[ eps (8 12 8 2 [

2 3 5 alpha y1

- ------------) 12

] ] ] ] ] ] ] ] ]

and transformation to polar coordinates using the program TRANSFORM gives:

2 dr

eps

3

2

r

- 4 eps

r

[[-- = - ------------------. dt

dtheta

8

dt

2 2 2 5 alpha eps r

= - ----------------]] 12

As an example of a nonautonomous system. we take the forced damped Duffing equation:

(28)

z

+

Z

=6

~

[

cos wt

- 0 z - a z

3

]

This equation is the natural symmetric extension of the forced damped harmonic oscillator and thus it occurs in a wide variety of applications. sample run of our program AVERAGE applied to eq.(28):

AVERAGE(); DO YOU WANT TO ENTER A NEW PROBLEM? (YIN)

Y; ARE YOU CX>NSIDERING A WEAKLY NONLINEAR OSCILLATOR OF THE FORM Z" + OMEGAO~2 Z = EPS F(Z.ZOOT.T) ? (YIN)

Y;

ENTER OMEGAO 1;

ENTER F(Z.ZooT.T) GAMMA*CX>S(W*T)-DELTA*ZDOT-ALPHA*Z 3; ENTER THE FORCING FREQUENCY A

W; ENTER THE ORDER OF THE SUBHARMONIC RESONANCE 1;

Here is a

AVERAGING

121

2 WE WRITE EPS*KAPOMEGA = w - 1 DO YOU WANT FIRST OR SECXlND ORDER AVERAGING?

(1/2)

2;

2 3 alpha y1 y2 gamma

2

matrix([eps

del ta gamma

(- ------------------- + ----------- + 4 3

16 w y1

3 alpha kapomega y2

2 delta y2

2

y2

4

8w

3 alpha y1

4

8w y2

kapomega y2

delta y1

--------)].

- -------------- +

2

2

8w

2w

2

2 w

8 w 2

[eps

4 256w

2

3

2

y2

2 kapomega y2

2 8 w

+ eps (-

y1

+ ------------)

- ----------------------- +

3 alpha y2

4

51 alpha

+ ----------------- - -------------------- + 4 4 128 w 8 w

3 alpha kapomega y1

4

2

3

y2

y2

256 w

8 w

223

51 alpha

5

51 alpha

2

3 alpha y2

gamma

9 alpha y1

gamma

kapomega gamma

(----------------- +

4

4 32 w

2

51 alpha

4

32 w

4

y1 y2

51 alpha

8 w

232 y1 y2

2

3 alpha kapomega y1 y2

----------------- +

4

4

2 51 alpha

4

128 w

256w

8 w

5 y1

2

3

3 alpha kapomega y1

delta

2

y1

kapomega -

- ------------- + 4 256 w

2

4 8 w

3 alpha y1 y2

+ eps (- ----- + 2 2 w

[VAX 8500 TIME

= 162

2 8 w

4 8 w

8 w

3

2

gamma

y1

------------)

delta y2

3 alpha y1

2 w

+ ----------2 8 w

kapomega y1

- -----------)]) 2

2w

SEC.]

These averaged equations may be simplified by transforming to polar coordinates (25) via the program TRANSFORM, as in the previous sample run.

AVERAGING

122

Neglecting terms of O(c2 ) for simplicity, we obtain:

dr [[-dt

-

(delta r w + sin(theta) gamma) eps 2 2 w

3

d the ta dt

(3 alpha r

- 4 kapomega r - 4 cos(theta) gamma) eps

=---------------------------------------------------- ]] 2 8 r w

An equilibrium of this averaged system corresponds to a limit cycle in

the original system (28). respectively for sin

a

Thus we set;

and cos

a ,

= e=0

in these last equations. solve

2 2 and use the identity sin a + cos a

=1

to

obtain an equation on the amplitude r of the limit cycle:

2

2

delta

r

2

w

3

2

(3 alpha r - 4 kapomega r) + ---------------------------2

2 gamma

=1

16 gamma

See Fig. 10. The purpose of our next example is to offer a check on our program by comparison with previously published second order averaging computations done by hand.

Holmes and Holmes [17] studied the stability and bifurcation of

subharmonic periodic solutions in the equation:

(29)

Note that when c u

=0

= 0,

~

cos wt - c 6 u

eq.(29) has equilibria at u

= 0,

!1.

The equilibrium at

is an unstable saddle, while those at +1 are stable centers. Let u

(30)

c

u

= U(t) u

be a periodic solution to (29) and set

U(t) + c Z

AVERAGING

123

5

r

o

-5 Fig. 10. Averaging O(e) solution for the steady state periodic response of the forced damped Duffing eq.(28). · d e. amp 1 Itu

U' sIng w2

for three cases:

~

\l

= de tuning ,

1 +" n e, we p 1ot

A (~

O. 6

= 0,

a

= 1)

B

(~

1. 6

= O.

a

= 1)

C

(~

1. 0

=

1. a = 1)

l'

= response

AVERAGING

124

where z is a variation from the periodic solution.

Substitution of (30) into

3 (29) gives to O(c ):

Eq.(31) governs the stability of. and bifurcations from. the solution u Holmes and Holmes [17] used regular perturbations to obtain an approximate expression for U(t), a periodic solution about the c equilibrium u

(32)

= 1: U(t)

where

where

f

Al

= 'Y/(2-w2 )

= -3f2 /4.

A2

2

faw/{2-w). A3

= -3f2 /(4-Sw2 )

Substitution of (32) into (31) gives

(33)

z

+ 2 z

c ( - 0

c

•

Z -

2

6f z cos wt - 3 z ) +

_? 2 2 fow. -(31- cos wt + 6 [-3f /4 + ----2 Sln wt

2 [

2-w

-

3~ cos wt]) z - 3f z 2 cos wt - z 3 2 4-8w

=0

= U(t).

AVERAGING

125

Holmes and Holmes [17] used second order averaging on eq.(33) to investigate the bifurcation of subharmonics of order 2. i.e. the phenomenon of period doubling.

We replicate their "tedious but elementary calculations" in

the following run of the program AVERAGE.

We begin by assigning the right-hand

side of eq.(33) to a variable called RHS: RHS:-DEL*ZDOT-6*GAM*COS(W*T)*Z-3*Z A2+ EPS*(-(3*GAMA2*COS(W*T)A2+6*(-3*GAM A2/4+GAM*DEL*W/(2-W A2)*SIN(W*T) -3*GAM A2/(4-8*W A2)*COS(2*W*T»)*Z-3*GAM*COS(W*T)*Z A2-ZA3); 2 3 gam cos(2 t w)

gam w sin(t w) del

(------------------

eps (- z (6

-

----------------2

2

2 cos (t w»

2

- ------) 4

4 - 8 w

2 - w

+ 3 gam

2 3 gam

2 2 - 3 gam cos(t w) z ) - zdot del - 3 z

3

- z

- 6 gam cos(t w) z AVERAGEO; DO YOU WANT TO ENTER A NEW PROBLEM? (YIN) Y; ARE YOU CONSIDERING A WEAKLY NONLINEAR OSCILLATOR OF THE FORM Z" + OMEGAOA2 Z = EPS F(Z.ZOOT.T) ? (YIN) Y;

ENTER OMEGAO SQRT(2) ; ENTER F(Z,ZOOT.T) RHS; ENTER THE FORCING FREQUENCY

W; ENTER THE ORDER OF THE SUBHARMONIC RESONANCE 2;

2 WE WRITE EPS*KAPOMEGA

=w

- 8 DO YOU WANT FIRST OR SECOND ORDER AVERAGING?

(112)

2'

2

y2 del (------2

2

matrix([eps

6 gam yl del

- ------------

2

2 2 w

y2

-

2

2 w 3 yl

6 gam y1 del

w (w

3

- 2)

w

3 3 60 y2 3 y2 - ----- + 2 4 2 w w

2 2 2 kapomega y2 3 gam kapomega y2 6 gam y2 60 yl y2 + --------- + --------- + ------------ + ----------------4 w

2 w

4 8 w

4 w

AVERAGING

126 2

9 gam

y1 del

y2

6 gam y2

kapomega y2

+ ---------) + eps (- ------ + ----------- + --------)], w 2 2 4 2 w w w

2

2

2

2 60 y1 y2

3 y1 y2

6 gam y2 del

6 gam y2 del

y1 del

3 3 y1

(- ------- + ------------ + ------------ + -------- - --------- + 4 2 3 2 2 2 2 w w 2 w 2 w w w (w - 2)

reps

3

60 y1

2 6 gam y1 w

y2 del + eps (-

kapomega y1 - ----------- +

2

w

2w [VAX 8500 TIME

3 gam kapomega y1

y1

= 271

9 gam -

- ------------ + 4 8 w

2

4 w

2

2

kapomega

y1

---------) 4 w

4

w

6 gam y1

--------)]) 2 w

SEC.]

These averaged equations agree with eq.(25) in [17] (but note that our definitions for t and 0 are different from those in [17].) The equilibria of the averaged equations correspond to periodic motions for the unaveraged system (33).

The averaged equations always possess an

equilibrium at the origin. corresponding to the solution u = U(t). additional equilibria correspond to subharmonics of order 2. order

t

Any

Note that the

terms in the averaged equations are linear in Y1 and Y2' and thus

cannot account for such additional equilibria.

Hence in order to investigate

2

period doubling, it is necessary to go to 0(6 ) in this example. reader to [17] for further analysis of this system.

We refer the

AVERAGING

127

Exercises

1.

One may envision bifurcation problems where second order averaging still

leads to a degenerate autonomous dynamical system. perform m-th order averaging would be welcome.

In such a case a program to

It is not too difficult to

write such a program bearing in mind that in order to find the m-th order mean gm(y) we have to find the transformation

2

x

y + e wI + e w2 + ... + e

m-I

wm- I

Sketch of a solution: One way to organize such a procedure is as follows:

Since we are

interested in the m-th order mean we can work with the Taylor expansion of eq.(3) up to order m:

m-I (PI)

2:

(-e)

i=O

i

aWl ] [ e fey + e wI,t,e) - e ---]

at

Performing first order averaging on (PI) gives gi and wI'

Substituting the

latter back into (PI) gives a new vector field f(y,t,e) which we now have to average to second order.

This time we determine the Taylor eXPansion to order

m of

and so on. below [2]:

A listing of the crucial portion of the function AVERAGE() is given

128

AVERAGING

JUMP 1 ,

1* AVERAGING *1

M:READ("ENTER ORDER OF AVERAGING"), IF M=1 1* FIRST ORDER AVERAGING *1

THEN (TEMPO:DEMOIVRE(APPLYl(VF2,KILLEXP». RESULT:TAYLOR(TEMPO.EPS,O.I» 1* AVERAGING OF ORDER M > 1 *1

ELSE Y:MAKELIST(CONCAT(·Y.I).I.I.N). WLIST:MAKELIST(CONCAT(·W.I),I.l,N). DEPENDS(WLIST.CONS(T.Y». TRAFO:Y. XSUB:MAPLIST("=". Y.X). 1* WNULL SETS WLIST TO ZERO *1

WNULL:VFUN(WLIST.O). JACOB[I.J] := DIFF(WLIST[I].Y[J]). JAC:GENMATRIX(JACOB.N.N). 1* MAIN LOOP *1

FOR I : 1 THRU M-l DO ( TEMPI :MAPLIST("=" .X. Y+EPSAI*WLIST). 1* HERE X IS THE LIST [Xl.X2 ..... XN]. Y IS THE LIST [Yl.Y2 ..... YN].

WLIST IS THE TRANSFORMATION VECTOR [WI. W2. , . , .WN] *1 TEMP2:TAYLOR(SUM(-EPS)A(I*J)*JAC J.J.O.M-l). AA

(EV(VF2.TEMPl) - DIFF(WLIST.T)*EPSAI).EPS.O.M). 1* JAC IS THE JACOBIAN D WLISTIDY OF THE TRANSFORMATION WLIST *1

TEMP3:MATTOLIST(TEMP2.N) . TEMP4:MAP(EXPAND.TAYLOR(EV(TEMP3,WNULL,DIFF),EPS,O.I». 1* THE ITH ORDER MEAN *1

AVERAGING

129

MEAN:APPLY1(TEMP4,KILLEXP), TEMP6:EXPAND(TEMP4-MEAN), TEMP7:EV(INTEGRATE(TEMP6, T) ,EPS=1). 1* THE ITH ORDER TRANSFORMATION *1

TEMP8:MAPLIST("=",WLIST,TEMP7), TEMP9:RATSIMP(TEMP8), 1* THE TRANSFORMED DE *1

YF2:EXPAND(EV(TEMP3,TEMP9,DIFF,XSUB,INFEVAL», 1* THE SUM OF ALL TRANSFORMATIONS *1

TRAFO:EXPAND(TRAFO+EV(EPS I*WLIST,TEMP9»), A

1* END OF MAIN LOOP *1

PRINT("THE TRANSFORMATION: ",X, "="), PRINT(RATSIMP(DEMOIYRE(TRAFO»). 1* THE FINAL AVERAGING *1

RESULT:APPLY1(YF2,KILLEXP), 1* REPLACE X BY Y *1

FOR 1:1 THRU N DO RESULT:SUBST(CXlNCAT('Y,I),CXlNCAT('X,I),RESULT), RESULT) $

1* AUXILIARY FUNCTIONS *1

YFUN(LIST, VALUE) :=MAP(LAMBDA([U], U=VALUE) ,LIST) $ MATTOLIST(MAT,DIM):=IF DIM>1 THEN MAKELIST(MAT[I,1],I,1,DIM) ELSE [MAT]$

For most practical purposes this program is probably too slow and creates intermediate expressions which are too large. one-dimensional problem like the following:

However it works for a

AVERAGING

130 AVERAGE() ;

00 YOU WANT TO ENTER A NEW PROBLEM? (YIN)

Y; ARE YOU CONSIDERING A WEAKLY NONLINEAR OSCILLATOR OF THE FORM + OMEGAO A2 Z = EPS F(Z,ZOOT,T) ? (YIN)

Z"

N;

ENTER NUMBER OF DIFFERENTIAL EQUATIONS

1; THE ODE' S ARE OF THE FORM: DX/DT = EPS F(X,T) WHERE X [xl] SCALE TIME T SUCH THAT AVERAGING OOCURS OVER 2 PI ENTER RHS( 1 )=EPS* ... (Xl-Xl A2)*SIN(T)A2;

=

D xl /DT

= eps

2

2

sin (t) (xl - xl )

ENTER ORDER OF AVERAGING

3; THE TRANSFORMATION:

[xl]

2 2 3 [- «2 eps cos(4 t) - S eps cos(2 t» yl

2 + (- 3 eps

2

2 + (eps

2

cos(4 t) - 16 eps sin(2 t) + 12 eps

cos(2

t»

yl

2 cos(4

t) +

eps

16 eps sin(2

t) -

4 eps

cos(2

343332 yl eps yl eps yl

t) -

64) yl)/64]

2 eps yl

eps yl

[- -------- + -------- - -------- - ------- + ------] 64 [VAX 8500 TIME = 48 SEC.]

32

64

2

2

AVERAGING 2.

131

Find a solution of the undamped Duffing equation «28) with 6

Lindstedt's method up to first order in c.

= 0)

via

Show that to this order the

amplitude of the periodic solution is identical to the steady state solution of the first order averaged system of (28) with 6

Hint:

= O.

We showed that the latter is of the form:

3 a r

3

4

- 4

n

r

=1

'Y

where r is the amplitude of the limit cycle and method to obtain YO(wt)

= A cos

wt, where w

'Y

2 A

+ 3 A2 8

=1

n = (w2-1)/c. + k

1

Use Lindstedt's

c, and where

OlAPTER 6

LIE TRANSFORMS

Introduction

The method of Lie transforms is a normal form method for Hamiltonian systems.

Like the method of normal forms (Chapter 3), Lie transforms involves

finding a change of variables so that the system of differential equations becomes simpler.

That is, the perturbation expansions are performed on the

transformation of coordinates rather than on the solution as a function of time (as, e.g., in Lindstedt's method.) The difference between Lie transforms and normal forms lies in the special nature of Hamiltonian systems.

Such systems occur extensively in

physics, engineering, astronomy and classical mechanics, and normally involve models which do not dissipate energy through damping.

The differential

equations for a Hamiltonian system are generated by a single scalar quantity, the Hamiltonian, which often can be identified with the total energy of the system.

(1. 1)

(1.2)

The equations take the form:

LIE TRANSFORMS

where H

~

133

H(qi,Pi,t) is the Hamiltonian, and where qi(t) and Pi(t) are the

dependent variables describing the state of the system.

For a system with n

degrees of freedom, i goes from 1 to n. As an example of a Hamiltonian system, let us take a simple harmonic oscillator.

In this case the Hamiltonian is:

(2)

whereupon the differential equations of motion become:

(3.1)

dq1 dt

(3.2)

(ft=

~

PI

dPl -

Cil

2

q1

where q1 corresponds to the displacement of the oscillator, and PI to its velocity.

(The q. and p. are traditionally called the generalized coordinates 1

1

and momenta, respectively.) In the method of Lie transforms, we are concerned with generating a change of coordinates which will simplify the Hamiltonian, whereas in the method of normal forms we deal directly with the differential equations of motion.

All quantities, including the original Hamiltonian H and the change of

coordinates, are expanded in a power series in a small parameter c:

(4)

H

The method generates a canonical coordinate transformation, i.e., it preserves the Hamiltonian form of the equations.

Thus the original system based on H

with variables qi,Pi is transformed into a new system with a new Hamiltonian K with variables Qi'P , i

The transformation itself is specified by a generating

LIE TRANSFORMS

134

function W.

Both K and Ware expanded in perturbation series:

+ •••

(5)

W

(6)

Up to terms of order en, the method gives that K depends upon W and other n n quantities which are already known.

The strategy of the method is to choose W n

so as to simplify K as much as possible. n

As usual in normal form methods, the

choice of the final canonical form K is strongly dependent on the specific details of the problem at hand. We shall now present the explicit details of the method. reader to [7J,[24J for a derivation of the following results. system with N degrees of freedom.

We refer the

We consider a

Let us denote the Poisson bracket of two

functions f(Q.,P.) and g(Q.,P.) by {f,g}: 1

1

1

1

N

(7)

{f,g}

1

af

ag

i=1 aQi aPi

_

£L ag aPi aQi

We define the operators L and Sn as follows: n

(8)

(9.1)

(9.2)

L

n

Id

(the identity operator)

L S, n n-m m

1,2,3, ...

Then the near-identity change of variables from (q.,p.) to (Q 1.. P.) is given by 1 1 1

LIE TRANSFORMS (10.1)

qi

[SO + E. SI + c

(10.2)

Pi

[SO + c SI + c

2

2

S2 +

135

... ] Qi

S2 + ••• ] P.1

and the nth term K of the resulting Hamiltonian K of eq.(5) is n

(11.1)

(11. 2)

(11.3)

K

n

H

n

+

1n

n~l [Ln-m Km + m Sn-m Hm],

[aWn + {W ,H )] + 1at n o n L m=l

n =

2,3,4 ..

in which the functions Hi' K and Wi are written as functions of the i transformed variables Q and Pi' i

By choosing W appropriately in eqs.(ll), we n

may simplify the expression for K . n

Action-angle Variables

In this Chapter we shall be interested in applying the method to systems which, when E.

= 0,

correspond to N linear oscillators, cf. eq.(2).

In

this case all the computations are greatly simplified by using a polar coordinate system called action-angle variables for the c

=0

problem.

In the

case of eq.(2), if we make the (canonical) transformation from ql,Pl to ¢l,J 1 via the equations (see Exercise 3 at the end of this Chapter)

(12.1)

(12.2)

LIE TRANSFORMS

136 we obtain

(13)

Here the coordinate ¢1 is called the angle and the momentum J

1

is the action.

Note that the eq.(13) is much simpler algebraically that the equivalent eq.(2). While the use of action-angle variables is not an essential part of the method of Lie transforms, we shall find it useful for the systems of forced, coupled nonlinear oscillators which we shall consider. As an example of the method. we consider the follOWing nonlinear Mathieu equation [30]:

(14)

d~

--- + (0 + c cos t) u + c a u

dt

2

3

0

which may be written in the Hamiltonian form (1) by setting

(15)

q

u,

p

du dt

with the Hamiltonian

(16)

H

2 2 4 (0 + c cos t) S- + caS2 2 4

=~ +

We shall set

(17)

where w is the frequency of the c parameters.

=0

problem and the 0i are detuning

The Hamiltonian may be simplified by using action-angle variables

LIE TRANSFORMS

¢. J

137

of eq. (12) :

(18)

H

=w

J

+ c

[° 1 + cos t] ~ sin2¢ + c

J:

a

4 2 sin ¢ + 0(c )

w

The near-identity transformation of variables (10) from ¢.J to. say. ~.I

coordinates is given by:

¢ =~ -

(19.1)

J

(19.2)

=I

aWl 2 c --- + O(c ) al

aWl 2 + c --- + O(c ) a~

where we have used, e.g.,

Sl~

= L1~ = {W1'~}

aWl

a~

a~

aWl

= ---- ---= a~ al aI a'/1

awl al

In the new coordinates. the terms of the transformed Hamiltonian are:

(20)

aWl

+ --+

(21. 1)

at

[°1 + cos

(21.2)

aw 1 I 2 12 . 4 t ] - sin '/1 + a :2 SIn ~ + - - + at

w

W

In order to choose WI judiciously, we

trigonometric~lly

aw 1

W --

a'/1

reduce the

right hand side of eq.(21.2):

(21.3)

K1

= 4~

[- cos(t+2'/1) -

12 [

cos(t-2~)

+!!..- cos 4'/1 - 4 cos 2~ + 3 2 8w

+ 2 cos t - 2

]

aWl

°1 cos 2'/1 + 2 0lJ

aWl a~

+ -- + w --

at

138

LIE TRANSFORMS

Each term on the right hand side of (21.3) which is of the form

(22)

A cos (B t + C

~)

may be removed by including a corresponding term in WI of the form:

D sin (B t + C

(23)

~)

which leads to the choice

(24)

WI

=I

sin(t+2~)

4w (1+2w)

I sin(t-2~) 4w (-I+2w)

2 sin 2,p I sin t + (ol Iw + aI ) - 43w 2 w

and thus yields the expression

(25)

Note that we do not remove the constant terms from K , as these would require a I term in WI that is linear in t or ,p, and such a term would destroy the uniform validity of the asymptotic expansion for WI as t or

~

goes to infinity.

The significance of eq.(25) is that now the transformed Hamiltonian K or t to order c:

does not involve

~

(26)

K = KO + Kl c

=w I

01 I

2 3aI ]

+ c [---- + --2-2w 8w

Thus the transformed problem is easy to solve:

LIE TRANSFORMS

139

(27.1)

dI dt

(27.2)

= _ oK = 0 o'i;

from which

(28.1)

(28.2)

By substituting (28) and (24) into (19) we may obtain expressions for the original action-angle variables original variable u

¢ and

= q = ~ sin ¢ is

J.

Then an expression for the

obtainable.

Computer Algebra

The key formulas (7)-(9) of the method of Lie transforms are ideally suited for implementation on MACSYMA [22].

We shall demonstrate a program

called LIE on the foregoing example. and then give the program listing. annotate the run in Italics:

LIE();

00 YOU WANT TO INPUT A NEW PROBLEM (YIN) ?

y; ENTER NUMBER OF DEGREES OF FREEOOM 1;

ENTER SYMBOL FOR Q[ 1 ] Q; ENTER SYMBOL FOR P[ 1 ]

P; THE HAMILTONIAN DEPENDS ON THE Q'S. P'S. T AND E (SMALL PARAMETER)

We

LIE TRANSFORMS

140

THE E=O PROBLEM MUST BE OF THE FORM: 2 2 w q

2 P

+

1

H

= ---------2

ENTER THE HAMILTONIAN PA2/2+(W 2+DELl*E+DEL2*EA2+E*COS(T»*QA2/2+E*A*QA4I4; A

2

2 2 (w + e cos(t) + de12 e + dell e)

q

H

4 2 a e q p + ------ + 4 2

= ------------------------------------2

THE ACfION-ANGLE VARIABLES ARE J'S FOR ACfION, PHI'S FOR ANGLE e cos(t + 2 phi)

j

1

H

j

1

1

e cos(t - 2 phi)

j

1

e cos(t)

j

1

W - -------------------- - -------------------- + -----------

4w

4w

2

2w

2

cos(2 phi ) del2 e

j

1

j

del2 e

j

cos(2 phi ) dell e

I I I I I - ---------------------- + ---------- - --------------------- + 2 w 2 w 2 w

2 j

2

cos(4 phi) a e

cos(2 phi ) a e 1 I

dell e

j

1

2w

2 3 j

a e I I I + ------------------ - ------------------ + -------j

2 2 2 8w 2w 8w

ENTER HIGHEST ORDER TERMS IN E TO BE KEPT I;

The program up to this point has involved entering data and transforming to action-angle variables iI' phil'

These preliminary steps

completed, the program begins the Lie transform computation.

The transformed

action-angle variables are denoted by iI' psi , and the first result displayed l is the generating function WI' referred to as WGEN[l] to avoid confusion with WI"

This equation corresponds to eq.(24) above:

LI E TRANSFORMS

141

WGEN[ 1 ] 2

i

sin(t + 2 psi) 1

1

__________________ L

2 8 w + 4 w

i

sin(t - 2 psi) I

I

2 8

w - 4 w

sin(2 psi) (i

dell w +

a)

1 1 + -----------------------------3 4 w

2

i

sin(t)

i

sin(4 psi) a

1

1 2 w

3

32 w Having obtained WI' the program computes K as in eq.(25) above: l THE TRANSFORMED HAMILTONIAN K[ I ] 2 i

dell

3 i

1 ------- + 2 w

a

1

2 8 w

The foregoing steps of computing Wi and K would now be iterated until i the specified truncation order is reached.

Since we specified truncation at

order c, the main loop is now exited, and the resulting expression for the transformed Hamiltonian K (called the Kamiltonian after Goldstein [12]) is displaued:

THE TRANSFORMED HAMILTONIAN IS 2 i

K

dell

3 i

a

1 1 i w + e (------- + ------) 2 2 w 1 8 w

The user is now given the option of generating the near-identity canonical transformation from jl' phil to iI' psi l action-angle coordinates:

LIE TRANSFORMS

142

DO YOU WANT TO SEE THE NEAR IDENTITY TRANSFORMATION (YIN) ?

Y; 2 i cos(t - 2 psi) cos(t + 2 psi) I I I I e (-------------------- + 2 2 8 w - 4 w 8 w + 4 w 2 i

j

I

2 2 cos(2 psi ) (i dell w + i a) i cos(4 psi) a I 1 I I I + ------------------------------ - ----------------) + i 3 I 3 2 w 8 w sin(t + 2 psi)

sin(t - 2 psi)

sin(2 psi ) (dell w + 2 i a) I I

I I e (- --------------- + --------------- -

phi

2

I

8w

2 +4w

8w

3 4w

-4w

i sin(4 psi) a sin(t) I 1 + ------ + ----------------) + psi 2 w 3 I 16 w

[VAX 8500 TIME = 52 SEC.]

Note that the O(e) computation wouLd not be vaLid for w

vanishing denominators.

due to

Such singuLar parameter vaLues are caLLed resonances.

If we had extended the computation to incLude e additionaL resonances w

= 1/2

=1

and w

= 1/4.

2

terms, we wouLd have found the

This behavior is typical of nonlinear

systems; as we extend the computation to higher and higher order in e, we Find more and more resonant parameter vaLues.

In order to examine typicaL behavior at a resonant parameter vaLue, we assign w the vaLue 1/4 (corresponding to 4:1 subharmonics) and again caLL the program LIE:

W:1I4$ LIE()$ DO YOU WANT TO INPur A NEW PROBLEM (YIN) ?

N;

LIE TRANSFORMS

143

By choosing not to input a new probLem, the previous HamiLtonian is used. subject to any parameter assignments that have been made since the program was Last run. We choose to truncate at c

2

terms in order to see the effect of the

resonance (which does not show up at order c):

ENTER HIGHEST ORDER TERMS IN E TO BE KEPT 2;

WGEN[ 1 ]

=

2 i

sin(t + 2 psi) 1 1 -------------------- + 2 i 3

1

sin(t - 2 psi ) - 2 i sin(t) 1 1 2

+ sin(2 psi) (4 i

1

THE TRANSFORMED HAMILTONIAN K[ 1 ]

2

dell + 16 i 1

a) - 2 i 1

sin(4 psi) a 1

1

=

2 2 i

dell + 6 i 1

a 1

WGEN[ 2 ] 2

4 i

sin(2 t + 2 psi )

a sin(t + 4 psi) 1 1 ---------------------- + ---------------------- - ----------------------15 3 3 1

4 i

1

sin(2 t - 2 psi ) 1 1

16 i

+ 5 additionaL Lines omitted for brevity

THE TRANSFORMED HAMILTONIAN K[ 2 ]

=

2 8 i

2 a cos(t - 4 psi) + 2 i

I

I

de12 - 8 i I

dell I

2

- 96 i

3

a dell - 272 i I

2

a I

4

i

1 + ----

3

144

LIE TRANSFORMS

THE TRANSFORMED HAMILTONIAN IS 2 K

e

2

2

a cos(t - 4 psi) + 2 i

(8 i

1

1

del2 - 8 i

dell

2

- 96 i

1

a dell 1

4 i 3

- 272 i

2 a

1

i

1 2 1 + ----) + (2 i dell + 6 i a) e + 3 1 1 4

As a check on the program. we note that this resuLt agrees with a previousLy pubLished hand caLcuLation [30J.

DO YOU WANT TO SEE THE NEAR IDENTITY TRANSFORMATION (YIN) ?

N; [VAX 8500 TIME

= 145

SEC.J

Note that in this resonant parameter case the transformed HamiLtonian K is not independent of angLe terms as it was for nonresonant w. cf. eq.(26). This is due to the unremovabiLity of terms with vanishing denominators. NevertheLess by making the canonicaL transformation from (29)

~

= ~1

~l.Il

to

~.I

via

- t/4 .

the new HamiLtonian becomes autonomous. and is therefore reLativeLy easy to anaLyze.

See Exercise 1 at the end of this Chapter.

Here is the program listing:

LIE(): =BLOCK(

1* INPUT PROBLEM ? *1

CHOICE1:READ("DO YOU WANT TO INPUT A NEW PROBLEM (YIN) ?"). IF CHOICEl=N THEN GO(JUMPl). 1* INPUT PROBLEM *1

N: READ ( "ENTER NUMBER OF DEGREES OF FREEDOM").

LIE TRANSFORMS FOR II: 1 THRU N 00 ( Q[II]:READ("ENTER SYMBOL FOR Q[",II,"J"), P[II] :READ("ENTER SYMBOL FOR P[", II, "J"». KILL(W) , PRINT("THE HAMILTONIAN DEPENDS ON THE Q'S. P'S, T AND E (SMALL PARAMETER)"). PRINT("THE E=O PROBLEM MUST BE OF THE FORM:"), PRINT("H =" . SUM ( (P[IIY2+W[IIY2*Q[IIY2)/2, II, 1 ,N», HORIGINAL:READ("ENTER THE HAMILTONIAN"). PRINT("H "," . HORIGINAL) ,

1* TRANSFORM TO ACTION-ANGLE VARIABLES *1 1* FIND THE W[II]'S *1 HO:EV(HORIGINAL,E=O), FOR II: 1 THRU N 00 W[II]:SQRT(DIFF(HO,Q[II],2», PRINT("THE ACTION-ANGLE VARIABLES ARE J'S FOR ACTION, PHI'S FOR ANGLE"), FOR II: 1 THRU N 00 TR[II]:[Q[II]=SQRT(2*J[II]IW[II])*SIN(PHI[II]), P[II]=SQRT(2*J[II]*W[II])*COS(PHI[II])], TRAN:MAKELIST(TR[II],II,l,N), H:EV(HORIGINAL, TRAN,ASSUMEJOS:TRUE, INFEVAL). H:TRIGSIMP(H), H:EXPAND(TRIGREDUCE(EXPAND(H»), PRINT("H =",H).

JUMPI.

1* INPUT TRUNCATION ORDER *1 NTRUNC:READ("ENTER HIGHEST ORDER TERMS IN E TO BE KEPT"), FOR I I: 0 THRU NTRUNC 00 H[II]:RATCOEF(H,E,II),

145

146

LIE TRANSFORMS

1* LIE TRANSFORMS *1 1* NEAR IDENTITY TRANSFORMATION FROM (J. PHI) .S TO (I. PSI) •S *1 1* UPDATE VARIABLES *1

FOR II: 1 THRU N OO( P[IIJ:I[IIJ. Q[II]:PSI[II]). 1* REPLACE J AND PHI BY I AND PSI IN H'S *1

FOR II: 0 THRU NTRUNC 00 H[II]:EV{H[IIJ.MAKELIST{J[IIIJ=I[III].III.I.N). MAKELIST(PHI[III]=PSI[III].III.I.N». K[OJ:H[O]. 1* DECLARE WGEN[I] TO BE A FN OF T. Q'S AND P'S *1

KILL(WGEN). DEPENDS(WGEN. [TJ). FOR II: 1 THRU N 00 DEPENDS(WGEN.[Q[II].P[IIJ]). 1* E=O PROBLEM IS OF FORM SUM(W[IIJ*I[IIJ) *1 1* CHOOSE WGEN[II] TO KILL AS MUCH AS POSSIBLE IN EQ(II) *1 1* EQUATE K[IIJ TO UNREMOVABLE TERMS *1 1* DEFINE PATTERN MATCHING RULES, TO ISOLATE ARGS OF TRIG TERMS *1

MATCHDECLARE{ [DUMMYI. DUMMY2J . TRUE) • DEFRULE(mSARG. DUMMYI*CXlS(DUMMY2) .DUMMY2). DEFRULE(SINARG.DUMMYI*SIN(DUMMY2).DUMMY2). FOR II: 1 THRU NTRUNC OO{ EQN[II]:EXPAND{TRIGREDUCE{EXPAND(EQ(II»». TEMP:EXPAND(EV(EQN[II].WGEN[II]=O». 1* CHANGE SUM TO A LIST *1

TEMPI: ARGS(TEMP).

LIE TRANSFORMS

/*

REMOVE

147

CONSTANT TERMS */

TEMP2:MAP(TRIGIDENTIFY,TEMPl) , /* ISOLATE ARGUMENTS OF TRIG TERMS */ ARG1:APPLY1 (TEMP2. COSARG. SINARG) . /*

REMOVE DUPLICATE ARGUMENTS

*/

ARG2:SETIFY(ARGl). /* REMOVE

RESONANT ARGUMENTS */

ARG3:SUBLIST(ARG2,NOTRESP), /* CHOOSE WGEN TO ELIMINATE NONRESONANT TERMS

*/

LENG:LENGTH(ARG3), WGENTEMPI :O. FOR JJ: 1 THRU LENG OO( WGENTEMP2:AAA*COS(PART(ARG3,JJ»+BBB*SIN(PART(ARG3,JJ». TEMP4:EV(EQN[II].WGEN[II]=WGENTEMP2,DIFF), TEMP5:S0LVE([RATCOEF(TEMP4,COS(PART(ARG3.JJ»). RATCOEF(TEMP4,SIN(PART(ARG3.JJ»)].[AAA,BBB]). WGENTEMPl:WGENTEMPl+EV(WGENTEMP2,TEMP5». WGEN[II]:WGENTEMPl, PRINT("WGEN[", II. "] ="). PRINT(WGEN[II]). K[II]:EXPAND(EV(EQN[II].DIFF», K[II]:EXPAND(RATSIMP(K[II]». PRINT("THE TRANSFORMED HAMILTONIAN K[" .II."] ="), PRINT(K[II]» , KAMILTONIAN:SUM(K[II]*EAII,II,O,NTRUNC), PRINT ("THE TRANSFORMED HAMILTONIAN IS "). PRINT ("K =" .KAMILTONIAN) , CHOICE2:READ("OO YOU WANT TO SEE THE NEAR IDENTITY TRANSFORMATION (Y/N) IF CHOICE2=N THEN GO(END).

?"),

LIE

148

TRANSFORMS

1* THE NEAR IDENTITY TRANSFORMATION *1

FOR II: 1 THRU N OO( JTRANS[II]:SUM(S(III.P[II])*EAIII.III,O,NTRUNC), PHITRANS[II]:SUM(S(III.Q[II])*EAIII.III,O,NTRUNC», FOR II: 1 THRU N OO( PRINT(J[II], "=", JTRANS[II]), PRINT(PHI[II]. "=", PHITRANS[II]»,

END,

KAMILTONIAN)$

1* AUXILIARY FUNCTIONS *1

POISSON(F, G) := SUM(DIFF(F,Q[II])*DIFF(G,P[II])-DIFF(F.P[II])*DIFF(G,Q[II]),II.l,N)$

L(II,F):=POISSON(WGEN[II].F)$

S(I1,F):=(IF 11=0 THEN F ELSE SUM(L(I1-M,S(M.F)).M,O,II-l)/II)$

EQ(II):=(H[II]+(DIFF(WGEN[II].T)+POISSON(WGEN[I1],H[O]»1 II +SUM(L(II-M,K[M])+M*S(II-M,H[M]),M,l,II-l)/II)$

LZAP(ANY):=DIFF(ANY,T)+POISSON(ANY.H[O])$

TRIGIDENTIFY(EXP):=IF FREEOF(SIN,EXP) AND FREEOF(OOS,EXP) THEN 0 ELSE EXP$

NOTRESP(EXP):=NOT IS(LZAP(EXP) = 0)$

LIE TRANSFORMS

149

SETIFY(LIST):=( SET: [LIST[l]], FOR I: 2 THRU LENGfH(LIST) DO( IF NOT MEMBER(LIST[I],SET) THEN SET:CONS(LIST[I] ,SET», SET) $

Example with Two Degrees of Freedom

In order to illustrate a two degree of freedom example, we take the problem of two identical nonlinear oscillators with nonlinear coupling [28],[29]:

This problem involves a resonance associated with the fact that the frequencies of the uncoupled linear oscillators are equal.

Here is the MACSYMA run of the

program LIE applied to this example:

LIEO; DO YOU WANT TO INPUT A NEW PROBLEM (YIN) ?

Y; ENTER NUMBER OF DEGREES OF FREEDOM 2;

ENTER SYMBOL FOR Q[ I ] QI; ENTER SYMBOL FOR P[ I ] PI; ENTER SYMBOL FOR Q[ 2 ] Q2;

ENTER SYMBOL FOR P[ 2 ] P2;

THE HAMILTONIAN DEPENDS ON THE Q'S, P'S, T AND E (SMALL PARAMETER) THE E=O PROBLEM MUST BE OF THE FORM: 2 w

2

2 + p2

q2

2

H = ------------ + 2

2

2

w ql 1

2 + pI

2

LIE TRANSFORMS

150 ENTER THE HAMILTONIAN

P1A2/2+Q1A2/2+P2A2/2+Q2A2/2+E*K*(Q1A4/4+Q2A4/4)+E*(Q1-Q2 )A4/4;

4 H

4

q2

=e

q1

2 q2

e (q1 - q2)

422 q1 p2

2 pI

k (--- + ---) + --- + ------------ + --- + --- + --4 4 4 2 2 2 2

THE ACTION-ANGLE VARIABLES ARE J' S FOR ACTION. PHI' S FOR ANGLE 2

2

cos(4 phi ) e k

j

2

H

2

= ------------------ -

j

2

cos(2 phi ) e k

j

cos(4 phi) e k

2 2 1 1 ------------------ + ------------------

8

2

8

+ 8 additional lines omitted for brevity

2 e 3 j e 121 + --------- + ------ + j + j 2 8 2 1 3 j

j

ENTER HIGHEST ORDER TERMS IN E TO BE KEPT 1-

WGEN[ 1 ]

=

2 2 sin(2 psi ) (i k + i + 3 i i ) 2 2 2 1 2

2 2 sin(4 psi ) (i k + i ) 2 2 2

4

32

+ 5 additional lines omitted for breVity

THE TRANSFORMED HAMILTONIAN K[ 1 ] 2

=

2

k 3 i i cos(2 psi - 2 psi ) 1 1 2 2 1 ------ + ------ + ---------------------------8 8 4

3 i

3 i

k

2

3/2 3 sqrt(i ) i cos(psi - psi) 12 21

3/2 3 i 1

2

sqrt(i) cos(psi - psi) 2 21

3 2

- -------------------------------- - -------------------------------- + ---2 2 8

2 3 i

i

1 2 + ------- + 2

3 i 1

8

LIE TRANSFORMS

151

THE TRANSFORMED HAMILTONIAN IS 2 3 i

K

=

2 k

3

k 3 i i cos(2 psi - 2 psi ) 2 1 1 2 2 1 e (------ + ------ + ---------------------------8 8 4

3/2 3/2 2 3 sqrt(i ) i cos (psi - psi) 3 i sqrt(i) cos(psi - psi) 3 i 12 21 1 2 21 2 - -------------------------------- - -------------------------------- + ----

2

2

8

2

3 i

3 i

i

1

2

1

+ ------- + ----) + i 2

8

+ i 2

1

DO YOU WANT TO SEE THE NEAR IDENTITY TRANSFORMATION (Y/N) ?

N; [VAX 8500 TIME

=

181 SEC.]

The resulting Kamiltonian.

can be simplified by the canonical transformation:

(32)

The resulting expression for K in terms of ~1.I1'~2.I2 does not involve ~2' ~

Thus 1

2

is a constant in time (since aK/a~2=O), and to O(e) the problem can be

reduced to that of a single degree of freedom system.

152

LIE TRANSFORMS

Exercises

1. For eq.(14) with w

= 1/4

in eq.(17). show that the Hamiltonian K which we

obtained.

'gives equations of motion which are equivalent to those corresponding to the new Hamiltonian

K(~.I)

under the transformation (29):

The equilibria of the corresponding equations of motion represent 4:1 subharmonic periodic motions of the original system (from eqs.(29).(12». Obtain an asymptotic expansion for the location of the equilibria of this 2 system. neglecting terms of 0(c ).

Using the fact that I must be non-negative.

show that there exists a bifurcation curve in the

o-c

plane given by:

(P2)

such that points lying to the left of this curve exhibit two 4:1 subharmonics. while points lying to the right exhibit no such subharmonics [30].

LIE TRANSFORMS 2. Consider eq.(14) with w

=

resulting Kamiltonian equations. the

From eq.(24) this choice of w

1/2 in eq.(17).

leads to a resonance in the order c terms.

153

Investigate the equilibria in the

Show that there are two bifurcation curves in

8-c plane given by:

(P3)

1

1

8=4"!2"c+···

which separate regions with O. 2. and 4 nontrivial equilibria. respectively correspond to O. 1. and 2

Show that these

2:1 subharmonic periodic motions

[18J.[30J.

3. A transformation is said to be canonical if it preserves the Hamiltonian

structure of the differential equations of motion.

The conditions for a

transformation from (q .• p.) to (Q .• P.) variables to be canonical is [3J: 1

1

1

1

(P4)

where the wedge product

(P5)

df

A

A

dg

is anticommutative:

- dg

A

df.

df

df

A

o

In computing the differentials dq. and dp. in (P4). we use the ordinary chain 1

1

rule based on the transformation of variables. linear combinations of the dQi's and dPi·s.

This gives dqi and dPi as

Show that the transformations

(12). (29) and (32) used in this Chapter are canonical.

LIE TRANSFORMS

154

4. Use Lie transforms to treat the undamped Duffing equation:

2

d x 3 --2 + (I + e A) x + e a x dt

(P6)

Take q

=x

and p

dx = dt and

=e

A cos t

use the Hamiltonian:

2 2 4

(P7)

H

=2 L + (1 +2 e A) L + e a S- 4

e A q cos t

Neglecting terms of O{e 2 ), find an approximate relationship between the amplitude of the periodic response and the parameters A, a and A.

CHAPTER 7

LIAPUNOV-SCHMIDT REDUCTION

Introduction

Like the center manifold reduction, the Liapunov-Schmidt reduction is a method which replaces a large and complicated set of equations by a simpler and smaller system which contains all the essential information concerning a bifurcation.

The method is applicable to a system of nonlinear evolution

equations of the form

dy

(1)

F(y,a)

dt

where a is a parameter.

A solution to (I), y

= y(t,a),

is said to exhibit a

bifurcation as a passes through a critical value a ' if the nature of the c solutions exhibits a qualitative change at a ' e.g. in the case of an c equilibrium solution. if the number of equilibria or their type of stability changes.

Here (1) describes either a finite dimensional or infinite

dimensional system.

The former we may think of a system of ordinary

differential equations, whereas the latter would be represented by one or more partial differential equations or integrodifferential equations. The aim of the Liapunov-Schmidt reduction is to reduce (I), which, even if finite dimensional, may be a very large system, to a small system of algebraic equations (typically one or t~o dimensional).

This will be

LIAPUNOV-SCHMIDT REDUCTION

156

accomplished by means of a perturbation expansion near a bifurcation point of a' known solution of (1).

It can be shown [13J,[42J that solutions of these

algebraic equations are locally in one to one correspondence to the solutions of (1).

An important restriction for our programs will be that we confine

ourselves to a nondegenerate bifurcation from a stationary solution to another stationary solution.

This excludes the important case of Hopf bifurcations,

which in principle one can treat with similar methods (cf.[13J).

Thus we shall

be interested in steady state solutions to the system (1):

F(y,a)

O.

Example: Euler Buckling

In order to introduce the method we perform an ad hoc perturbation analysis of the buckling problem. section.

We will generalize our approach in the next

A model for static beam buckling under compression (Euler's elastica,

[25J) is given by

(2)

+ ex sin y

o ,

y'(O)

= y'(l)

o

where x is the arc length along the deformed beam, y(x) is the angle which the tangent to the beam makes with the undeformed beam axis, ex is the load parameter and where the beam's length is normalized to 1. conditions correspond to pinned ends.

Eq.(2) has an analytical solution in

terms of elliptic integrals (cf.[40J)

(3)

ex

The boundary

m

1,2,3, ..

LIAPUNOV-SCHMIDT REDUCTION where

K(k)

=f

v/2

a

2 2 -1/2 (l-k sin ~) d~

first kind and k

sin a/2 where a

157

is the complete elliptic integral of the

= yeO)

is the beam angle at x

= O.

If we linearize (2) we have the equation for an harmonic oscillator

(4)

+ a y

Note that

y

2 2 a # n v . n

=a is

=0

•

y'

(0)

= 1.2.3.···.

= a c = v2

a

the only solution satisfying the boundary conditions if If a

= n 2v 2

then y

solution with undetermined amplitude A. Then a

y' (1)

= A cos

nvx is an admissible

Let us assume we increase a from O.

is the first value where a nontrivial solution may branch off

from the solution y

=O.

In order to find the deflection amplitude A as a

function of a parameter A = a - a ' we make the following ansatz: c

(5)

y

= A cos

vx + W(X;A.A)

where A is assumed to be small and where wand A are of second order in A.

We

require that a Fourier decomposition of W(X;A.A) has no component cos vx. since if it had we could include it in the A cos wx term.

Expanding (2) up to third

order we find

3 + a (y - ~ )

3!

(A cos vx + w) + (A + v 2 ) [A cos wx +

3 W _

(A cos wx + w) ] 3!

3

(6)

A A cos wx +

(A + v 2 ) [w _ (A cos ~ + w) ] 3!

a

LIAPUNOV-SCHMIDT REDUCTION

158

Since {cos nwx} is a complete orthogonal set on [0.1]. we can split the Fourier decomposition of (6) into two parts:

The cos wx component and all other modes.

To obtain the coefficient of the cos wx component of (6). we take the inner

1 product of (6) with cos wx and divide it by f

o

cos2 wx

dx = 1/2.

Since w. and

hence w", have. by hypothesis, no cos wx components. we obtain (after some trigonometric simplification):

(7.1)

"2

] ~ 1[3 A2 w cos 3wx + 3A w2 (1 + cos 2rrx) + 2w3 cos me

dx

]

0

All the other modes may be accounted for by subtracting the cos wx component (the coefficient of which is given by Eq(7.1)) from (6):

(7.2)

2 - (A + v ) [1_ A3 cos 3me + _3 A2 w (1+ cos 2wx) + 3A w2 cos me + w3 6 4 2

- cos me /

o

[

~ A2w cos

2 3 3me + 3A w (1 + cos 2me) + 2w cos me

2

We will later recognize eq. (7.1) as the bifurcation equation.

Jdx]

0

Note that to

solve (7.1) we have to know wand hence we must first solve (7.2).

A closer

look at (7.2) reveals that we are dealing with a forced nonlinear detuned oscillator.

Since w is assumed to be of second order in A and A. eq.(7;2)

reduces, for small A and A. to a linear forced harmonic oscillator

LIAPUNOV-SCHMIDT REDUCTION

159

(8)

The solution of the unforced system is given by

w

=B

cos rrx + C sin rrx.

A

particular solution to (8) is given by

3

(9)

w

A _ cos 3rrx - _ 192

and hence, in order to fit the boundary conditions w'(O) set C

=0.

cos rrx. 3 A

= w'(l) = 0,

we must

Morever B must be zero since w must not have a Fourier component

Thus (9) represents the solution w of (7.2).

Therefore up to order

the integrated terms in eq.(7.l) are negligible and we calculate the

response function A(A) to be a solution of

5 However, if we include terms in eq.(2) up to order A • and then determine the Fourier coefficient of the cos rrx mode and insert (9) into the resulting equation. we find:

~ A3 + 3

(10)

8

2 1T

A5

=0

512

We solve (10) for A and expand the result up to fourth order:

(11)

A

= 1T2

~

2 A +

8

~

4 A

512

A Taylor expansion of the elliptic integral in the exact solution (3) with respect to a. the beam angle at x

=0

• gives

LIAPUNOV-SCHMIDT REDUCTION

160

(12)

a

If we take into account that a a

A+

~

2

= yeO) = A

w(O;A,O)

+

=A

3

- A + ••• 192

and that

, then (11) and (12) are identical up to order 4.

Formal Setup

In the previous example we made several assumptions about the size of 2

small terms, e.g. wand A were assumed to be 0(A ).

These assumptions are, as

we will show, both unnecessary and, from the point of view of an automated computer algebra program, undesirable.

In this section we shall generalize the

perturbation analysis using the language of functional analysis in order to prepare for the approach used in the MACSYMA program.

We assume a physical

system whose dynamics are governed by the evolution equation (I), where y is an element of an appropriate Banach space E and y

=0

is a trivial solution.

E is

2

typically a subspace of the Hilbert space L (0) where 0 is a bounded domain in

rnn ,

furnished with the standard inner product



(13)

f u(x) ·v(x) dx

o

where u(x) and vex) are n-vectors whose elements are functions of x, and where u(x)·v(x) is the scalar product of u and v in

rnn .

If D F(O,a), the linear part of the operator F (i.e. its Frechet y

derivative), has only eigenvalues with negative real parts, then every perturbation of y

=0

in (1) is damped and will die away.

However if we have

an eigenvalue of L which is zero at a = a ' then the situation is marginal and c signifies the onset of branching at the bifurcation point A define the linear operator L to be D F(O,a). y c

=a

- ac

= O.

We

Since we only want to deal with

LIAPUNOV-SCHMIDT REDUCTION nondegenerate bifurcations. the eigenvalues will satisfy LY l

=0

= ~nYn

and LYn

~n

where Re

161

and the eigenfunctions Y of L n

~n<

O.

We call Yl the critical

eigenfunction and define the adjoint critical eigenfunction via L*Y* l where L* is the adjoint operator of L, i.e., we can decompose the Banach space E

A = 0 such that we can write Y "ampli tude" and Y1

Eo

N,

W Eo

R.

=N +

= A Y1 +

= .

=0

We assume that

R into kernel N and range R of L at

w, where A Eo ffi is a "state variable" or

=0

Thus Ly1

examples this splitting is always possible.

and w

= Lu

for some u.

In all our

Let us define projections onto the

subspaces Nand R:

PY

(14.1 )

(14.2)

Q

1 Id - P. i.e. , Q Y = Y - - - -

P projects onto the kernel N of L and Q onto the image R of L. P Y1

= Yl

and Q Yl

= O.

Since by definition L*Y*l

of the range of L we find that

(14.2) we find that Q w F(y,a)

(15)

= 0,

= w.

=

(y* 1 ,Lu>

= o and *

Observe that

since w is an element

= O.

Hence from

Next we replace the parameter a by A = a - a c in

and we write it in the form

F(y,A)

L

y +

F(y,A)

o

Thus we can split (15) into two equations on the subspaces N and R:

LIAPUNOV-SCHMIDT REDUCTION

162

(16.2)

o

PF

(16.1)

QF

= Q [L (A Yl

o

+ w) + F(A Yl + W,A)J

By this splitting we restrict L to operate on the complement of the kernel of L and hence L is invertible on R. W(X;A,A) to (16.2).

Inserting this into (16.1) we get a simple algebraic

=0

equation g(A,A,W(x;A,A» bifurcation equation. projection P F

=0

Therefore we can find a unique solution

(or simply g(A.A)

= 0)

which is the so-called

In our previous example. eq.(7.1) corresponds to the

and eq.(7.2) to Q F

= O.

Since W(X;A.A) can rarely be found

analytically in closed form as a solution of (16.2). one determines W(X;A.A) as a Taylor series in A and A to a fixed order. truncated Taylor series for g(A,A).

This leads consequently to a

Note that in order to determine g(A.A) to

order n, we need W(X;A,A) only to order n-1, since w enters (16.1) nonlinearly in y and A.

This gives us the nice possiblity of determining g to order n

without solving (16.2) if we know a priori (e.g. from symmetry arguments) that w is of order n.

We illustrate the method by reworking the elastica problem.

Euler Buckling Revisited

In this section we will redo the elastica problem in the way abstractly described in the previous section.

The method we follow here will be used in

our computer algebra implementation of the Liapunov-Schmidt reduction.

Let us

recall the differential equation for beam buckling

2

(17)

d y + a sin y O .

y'(O)

= y'(1) = 0

~ Based on our previous analysis. we take Yl

= cos

HX

as critical eigenfunction

LIAPUNOV-SCHMIDT REDUCTION

163

and Yl* = Yl as adjoint critical eigenfunction at a = a c = ~2 We define A = a - v 2 and substitute y = A cos vx + W(X;A,A) into (17) to give

(18)

-

2

~

2 A cos vx + d w + (A + v 2 ) sin(A cos vx + w)

=0

2

dx

2

Our linear operator L is given by ~ + v 2 and since we bifurcate from the dx2 trivial solution y = 0 we have

(19)

Y\A=A=O = w(x;O,O) = 0

Since g(A,A) and W(X;A,A) are polynomials in A and A, we can calculate them via Taylor expansions of equations (16).

The bifurcation equation g(A,A)

of (16.1) is obtained by taking the inner product of eq.(18) with Yl: *

1

(20)

g(A,A)

r = JCos o

2 2. ] dx vx [- 2 v A cos vx + ddxw 2 + (A + v ) sln(A cos vx + w)

= - V:A +

f~os

2

2 vx [ d ; + (A + v ) sin(A cos vx + w) ] dx

o

dx

Eq.(16.2) is obtained by operating with Q on (18), i.e., by subtracting

* g(A,A) Yl/

=2

g(A,A) cos vx from (18) (cf. (14.2», which gives

(21)

- 2 cos vx

f~os vx o

2 2 [ d ; + (A + v ) sin(A cos vx + w) ] dx dx

=0

LIAPUNOV-SCHMIDT REDUCTION

164

We will now obtain the first few terms of the Taylor expansion of

g(A.~).

We

use subscripts to represent the partial derivatives of g and w evaluated at A = 0 and

~

= O. e.g. gA = dgl

.

We begin by setting A =

d~ A=~=O

~

= 0 in (20) and

using (19) to obtain:

(22)

g(O,O)

=0

Next we differentiate (20) with respect to A

l

=

r~os

i

set A = A

11X

3 [d ;

dx~

= O.

+ sin(A cos

11X

+ w) +

(~

2

+ rr ) dw cos(A cos

~

11X

+ w) ] dx.

and use (19) to obtain

(23)

2 . d dw 2 dw dw The last equali ty holds S1nce - - - + rr - = L - and 2 dx dA d~ d~

* dw (L*Y1'~ * dw> = 0 since L* Y1* = O. (Yl,L d~> =

This result may be obtained directly

from (23) by using integration by parts. Next we differentiate (20) with respect to A.

dg dA

set A

2

= - 2rr

=~

+

IIo

cos

11X

[

3 d2 w - + (~ + rr2 ) (cos dxdA

O. and use (19) to obtain

11X

d + .2!.) cos(A cos dA

11X

+ w) ] dx.

LIAPUNOV-SCHMIDT REDUCTION _ ;2 +

(24)

J~OS

3 me [d ;

o

+

lr

2

dx dA

cos me +

165

lr

2

dw ] dx dA

I

A==A==O

The last equality holds by the same reasoning as used in deriving (23). Next we take the mixed partial derivative of (20) with respect to A and A. set A == A == O. and use (19) to obtain

==

J~os

me [ cos me + wA ] dx

o

where we have again used L** Y == O. 1 know w . A

We see that in order to find gAA we need to

This is accomplished via eq.(21). the Q-projection of eq.(18).

We

differentiate (21) with respect to A. set A == A == O. and use (19) to obtain

(26)

- 2 cos me

J~os o

2 me [ d w + 2 A dx

lr

2

(cos me + wA) ] dx == 0

The integral in (26) can again be simplified by integrating by parts and using L*Y* == O. giving the simple result 1

LIAPUNOV-SCHMIDT REDUCTION

166

a

(27)

Hence we have to solve L w A

=0

for wA"

However since w

6

R and since L is

invertible on R, we can conclude that

=0

(28)

For our specific case this can be seen by noting that L q

2 d q +

2

11"

q

o with

2 dx

q'(O)

= q'(I) = 0

eigenfunction Yl'

has the solution q

= cos

vx, which is the critical

Since we defined w to lie in the complement of the critical

eigenfunction. we arrive at the desired conclusion.

Therefore, from (25),

(29)

In this manner one can determine g as a polynomial in A and A up to any desired order.

It turns out that the first nontrivial coefficient for w is

given by w ' AAA

To find it, we differentiate (21) three times with respect to

A, set A

= A = 0,

(30)

d 2

and use (19) and (28) to obtain

2

dx

w + AAA

2

11"

wAAA -

2

11"

3

cos lI"X + 2 cos vx

l 2

Jo

11"

o

which can be more concisely written as

L w AAA

(31)

2

11"

3 Q(cos lI"X)

o

We write cos 3 vx as _1 (cos 3l1"X + 3 cos lI"X) and note that for this bifurcation 4

problem the projection Q can be performed by simply deleting the critical

LIAPUNOV-SCHMIDT REDUCTION eigenmode in the nonhomogeneous part of (31).

167

This leaves us with

2

L w AAA

= 7T__

cos 37TX.

Taking into account that the homogeneous equation has

4

zero as a solution in R. we find that

1 cos 37TX - __ 32

(32)

Computer Algebra

The MACSYMA program REDUCTION1 which we present in this section applies the Liapunov-Schmidt method to the problem of steady state bifurcations governed by a single nonlinear differential equation of the form

(33)

where the linear differential operator is required to be of the form

(34)

L

+ constant

We assume that the differential equation is defined on a one-dimensional space interval with either zero or zero-flux boundary conditions.

Note that if we

answer "N" to the question of whether some Taylor coefficients are known to be zero a priori. the program runs fully automatically.

However we may reduce the

execution time dramatically by setting to zero those coefficients of g and w which vanish for symmetry reasons.

We assume that y

=0

is a trivial solution

of the differential equation for all a. and that the bifurcation parameter a enters the problem only linearly such that no terms of the form AMpILAMJ for

J

>1

exist in g and w.

LIAPUNOV-SCHMIDT REDUCTION

168

Here is a sample run on the Euler buckling problem.

Since this problem

is invariant when y is changed to - y. all odd order derivatives with respect to amplitude A (= AMP) in wand g are zero: REDUCfIONI (); ENTER DEPENDENT VARIABLE Y;

USE X AS THE INDEPENDENT VARIABLE AND ALPHA AS A PARAMETER TO VARY ENTER THE CRITICAL BIFURCATION VALUE ALPHA %PI 2; A

2

WE DEFINE LAM = ALPHA -

%PI ENTER THE CRITICAL EIGENFUNCfION COS(%PI*X) ; WHAT IS THE LENGTH OF THE X-INTERVAL 1;

SPECIFY THE BOUNDARY CONDITIONS YOUR CHOICE FOR THE B.C. ON Y AT X=O AND X= 1 ENTER 1 FOR Y=O. 2 FOR Y'=O B.C. AT O? 2;

B.C. AT 1 ? 2;

THE D.E. IS OF THE FORM Y"

+

F(Y,Y' .ALPHA) = O. ENTER F

ALPHA*SIN(y) ; 2 d Y --- + 2

2

(LAM + %PI ) SIN(Y)

dX DO YOU KNOW A PRIORI THAT SOME TAYLOR COEFFICIENTS ARE ZERO. YIN Y; TO WHICH ORDER DO YOU WANT TO CALCULATE 5;

IS DIFF(W, AMP. 2, LAM. 0) IDENTICALLY ZERO. YIN Y; IS DIFF(W.AMP. 3. LAM. 0) IDENTICALLY ZERO. YIN N; 3

d W

COS(3 %PI X)

[----- = - ------------] 3

32

dAMP Compare this resuLt to eqs.(9) and (31).

LIAPUNOV-SCHMIDT REDUCTION IS DIFF(W.AMP. 4. LAM. 0) IDENTICALLY ZERO, YIN

Y; IS DIFF(W , AMP , 1. LAM, 1) IDENTICALLY ZERO. YIN

N; 2 d W [---------

OJ

:0

dAMP dLAM IS DIFF(W, AMP, 2. LAM, 1) IDENTICALLY ZERO. YIN

Y; IS DIFF(W.AMP. 3 LAM. 1

IDENTICALLY ZERO, YIN

N· 4 d W [----------

9 COS(3 %PI X) :0

-

3 dAMP IS Y; IS Y; IS N; IS Y; IS N; IS N; IS Y; IS N; IS Y;

--------------J 2

dLAM

256 %PI

GYOLY( 1,0 ) IDENTICALLY ZERO.

YIN

GYOLY( 2.0 ) IDENTICALLY ZERO,

YIN

GJQLY( 3.0 ) IDENTICALLY ZERO.

YIN

GYOLY( 4.0 ) IDENTICALLY ZERO.

YIN

GJQLY( 5.0 ) IDENTICALLY ZERO. YIN GYOLY( 1.1 ) IDENTICALLY ZERO.

YIN

GJQLY( 2,1 ) IDENTICALLY ZERO.

YIN

GYOLY( 3.1 ) IDENTICALLY ZERO. YIN GJQLY( 4,1 ) IDENTICALLY ZERO.

YIN

2 5 2 3 3 %PI AMP AMP LAM 3 %PI AMP AMP LAM - -------- + ------- + ----------- - --------16 2 1024 16 [VAX 8500 TIME

:0

390 SEC.J

This is the biFurcation equation.

We manipulate this result to be able to

compare with our introductory calculations:

SOLVE(%.LAM) ;

2 3 %PI

[LAM

4 AMP

2 - 64 %PI

2 AMP

= --------------------------J 2

64 AMP

- 512

LIAPUNOV-SCHMIDT REDUCTION

170

TAYLOR(%.AMP.O.4); 2

2

2

4

%PI AMP (5 %PI ) AMP [LAM + . . . = --------- + ------------- + . . . J 8 512

ITI

This resuLt agrees with equation (11).

Here is a listing of the MACSYMA functions:

1* This file contains REDUCTION1(), a function to perform a Liapunov-Schmidt

reduction for steady state bifurcations of nonlinear d.e.'s of the form Y"

+ F(Y,Y',ALPHA) = O.

Y = Y(X) is defined on a real interval with Dirichlet

or Neumann boundary conditions and F depends only linearly on ALPHA. It also contains these additional functions: SETUP() allows the problem to be entered. G-PQLY(I.J) calculates the coefficient of

AMP~I LAM~J

in the bifurcation

equation. W-PQLY(I.J) calculates the coefficient of AMpAI

LAM~J

in W(X;AMP,LAM).

PROJECT(EXP) ensures that =O. NEUMANNBC(EXP) accounts for Neumann boundary conditions. G-EQ() assembles the bifurcation equation. *1

REDUCTION1():=BLOCK( SETUP() • ORDER:READ(''TO WHICH ORDER DO YOU WANT TO CALCULATE"). FOR 1:2 THRU ORDER-1 DO W-PQLY(1,O), FOR 1:1 THRU ORDER-2 DO WJOLY(I.1). FOR 1:1 THRU ORDER DO GJOLY(I,O), FOR I: 1 THRU ORDER-1 DO G-PQLY (1, 1) . G-EQ() )$

LIAPUNOV-SCHMIDT REDUCTION

171

SETUP{) :=C /* The function SETUP asks for the variables of the d.e., the bifurcation

point, the critical eigenfunction. the x-interval. the boundary conditions and the differential equation. */

ASSUME..YOS: TRUE. LSJ.,IST: [J. Y:READC"ENTER DEPENDENT VARIABLE"), PRINTC"USE X AS THE INDEPENDENT VARIABLE AND ALPHA AS A PARAMETER TO VARY"), CAL:READC"ENTER THE CRITICAL BIF1JRCATION VALUE ALPHA"). PRINTC"WE DEFINE LAM = ALPHA - ".CAL). CFUN: READC"ENTER THE CRITICAL EIGENFUNCTION"), DEPENDSC[ZW,Y.W].[AMP,LAM.X]). LEN:READC"WHAT IS THE LENGTH OF THE X-INTERVAL"). NORM: INTEGRATECCFUN A 2.X.O.LEN) , PRINTC"SPECIFY THE BOUNDARY CDNDITIONS"), PRINTC"YOUR CHOICE FOR THE B.C. ON Y AT x=o AND X=",LEN) , PRINTC"ENTER 1 FOR Y=O, 2 FOR Y'=O"), BCL:READC"B.C. AT O?"). BCR:READC"B.C. AT".LEN,"?"). EQ:DIFFCY.X.2) + READC"THE D.E. IS OF THE FORM Y"

+ FCY, Y' •ALPHA) = O. ENTER F").

EQLAM:EVCEQ,ALPHA=LAM+CAL) , PRINTCEQLAM) , DATABASE: [DIFFCW, AMP)=O,DIFFCW. LAM) =0] , SUB: Y=AMP*CFUN+W • TEMPl:EVCEQLAM.SUB.DIFF). NULLANS: READC"OO YOU KNOW A PRIORI THAT SOME TAYLOR CDEFFICIENTS ARE ZERO, Y/N") )$

LIAPUNOV-SCHMIDT REDUCTION

172 G-POLY(I.J):=BLOCK(

1* This is a function to determine a particular Taylor coefficient of the

bifurcation equation G(AMP.LAM) =0. coefficients of W(AMP,LAM).

It requires knowledge about the Taylor

This knowledge is stored in the list DATABASE. */

LS-LIST:CONS{[K=I,L=J],LS-LIST), IF NULLANS = Y THEN ( ZEROANS:READ("IS G-POLY(". I,",",J.") IDENTICALLY ZERO. YIN"). IF ZEROANS = Y THEN RETURN(BIFEQ[I.J]:O», TEMP2:DIFF(TEMP1.AMP.I,LAM.J). DERIVSUBST:TRUE, 1* This derivative of w will be annihilated by the projection onto the critical

eigenfunction. hence we set it to zero here. */ TEMP3:SUBST('DIFF(W.AMP.I,LAM,J)=0.TEMP2). 1* Here we insert all knowledge gained through W-POLY *1 D....BASE-LENGTH:LENGTH(DATABASE) . FOR I I THRU D....BASE-LENGTH DO TEMP3:EV(SUBS(DATABASE[D....BASE-LENGTH+1-II].TEMP3).DIFF). DERIVSUBST:FALSE. TEMP4:EV(TEMP3,AMP=O.LAM=O.W=0). 1* Projection onto CFUN. the critical eigenfunction. *1 TEMP5:TRIGREDUCE(CFUN*TEMP4) , BIFEQ[I,J]:RATSIMP(INTEGRATE(TEMP5.X.0,LEN» )$

W-PQLY{I.J):=BLOCK( 1* This function allows the iterative determination of any particular Taylor

coefficient of the function W(AMP.LAM).

The result is fed into DATABASE. *1

IF NULLANS = Y THEN ( ZEROANS:READ("IS DIFF(W.AMP.",I.", LAM,".J,") IDENTICALLY ZERO .YIN").

LIAPUNOV-SCHMIDT REDUCTION

173

IF ZEROANS = Y THEN (ADDBASE:['DIFF(W,AMP, I ,LAM,J)=O] , DATABASE: APPEND(DATABASE.ADDBASE), RETURN(ADDBASE»), TEMP2:DIFF{TEMP1,AMP.I,LAM,J), DERIVSUBST:TRUE, 1* We substitute ZW for the unknown Taylor coefficient and solve an o.d.e. for

ZW in TEMP7 *1 TEMP3:SUBST(DIFF(W,AMP,I.LAM,J)=ZW,TEMP2), 1* Now we insert all knowledge gained through previous iterations. *1

D...J3ASEJ..ENGTH: LENGTH (DATABASE) , FOR II THRU D-BASEJ..ENGTH DO TEMP3:EV(SUBST(DATABASE[D-BASEJ..ENGTH+I-II].TEMP3),DIFF), DERIVSUBST:FALSE, TEMP4:EV(TEMP3,AMP=O,LAM=O.W=O). TEMP5:TRIGREDUCE(TEMP4) , 1* This ensures that the r.h.s. of the d.e. TEMP6 is orthogonal to the solution

of the homogeneous equation. *1 TEMP6:PROJECT(TEMP5). TEMP7: ODE2(TEMP6, ZW, X) , 1* Satisfy boundary conditions *1

IF BCL*BCR=1 THEN TEMP8:BC2(TEMP7,X=O,ZW=O,X=LEN,ZW=O) ELSE TEMP8 :NEUMANNBC{TEMP7) , 1* Make sure that =O *1

TEMP9:PROJECT(TEMP8), ADDBASE:['DIFF(W,AMP.I,LAM.J)=RHS(TEMP9)]. DATABASE:APPEND(DATABASE.ADDBASE). PRINT(ADDBASE»$

LIAPUNOV-SCHMIDT REDUCTION

174

PROJECT(EXP):=( PR01:EV(EXP,ZW=O), PR02:INTEGRATE(PR01*CFUN,X,O,LEN)/NORM, EXPAND(EXP-PR02*CFUN))$

NEUMANNBC(EXP) : =( REXP:RHS(EXP) , NBC1:DIFF(REXP,X), IF BCL=l AND BCR=2 THEN NBC2:SOLVE([EV(REXP ,X=O) ,EV(NBC1,X=LEN)J, [%K1,%K2J), IF BCL=2 AND BCR=l THEN NBC2:S0LVE([EV(REXP,X=LEN) ,EV(NBC1,X=O)J, [%K1,%K2J), IF BCL=2 AND BCR=2 THEN NBC2:S0LVE([EV(NBC1,X=LEN) ,EV(NBC1,X=O)J, [%K1,%K2]), EV(EXP ,NBC2))$

SUM(EV(AMp A K*LAMA LIK!*BIFEQ[K,LJ,LS-LIST[NJ),N,l,LENGTH(LS-LIST))$

The following theoretical results have gone into the design of the functions G-PQLY(I,J) and W-PQLY(I,J): (i) We want to study the bifurcation from y = 0 at A = O.

With

y(A,A) = A Y1+ W(X;A,A) and y(O,O) = 0 we have that w(x;O,O) = O. ~

(ii) From (16) we see that Q F(A Y1 + W,A) does not contain a simple linear term in A Yr'

Hence L wA = 0 and since L is invertible on R, wA = O. This is the first entry into the list DATABASE. Since we assume that y 0 is

=

a trivial solution, we have that F(O,O,a) = 0 and hence from differentiating (16.2) with respect to A and evaluating at A = A = 0, we find that wA This is the second entry into DATABASE.

=O.

LIAPUNOV-SCHMIDT REDUCTION

175

(iii) Eq(16.1) does not contain the term L w since P L w obtained in G-PQLY() at the stage TEMP3 where we set

di+jw

= O.

This is

0, since this

dAidA j term can only occur linearly (cf. the argument leading to Eq.(23)). (iv) The same term

di+jw

is the unknown variable in W-POLY for the

dAidA j Taylor coefficient AiA

j

in W(A,A).

We replace it by the variables ZW in TEMP3

of WJOLYO. (v) The projection Q is performed by the function PROJECT(EXP). Since the linear problem is self-adjoint, the projection is given by

LEN

J exp CFUN dx

Q(exp)

= exp

LEN

with NORM

=J

o

o

- . : . . - - - - - - CFUN NORM

2

CFUN dx.

This ensures that the ordinary differential equation

TEMP6 has a nontrivial particular solution.

The ode-solver in MACSYMA, ODE2,

adds the solution of the homogeneous equation to any particular solution it finds.

After having fitted the boundary conditions we then have to perform the

Q-projection again.

This ensures that the projection of w onto the critical

eigenfunction is always zero.

For the Euler buckling problem (indeed

for all problems with Neumann boundary conditions at both ends of the defining interval), this final projection is trivial and amounts to deleting the term which contains the critical eigenfunction explicitly.

However, in some cases

(e.g. quadratic nonlinearities with Dirichlet boundary conditions) we encounter resonance terms which make the projection Q nontrivial (cf. Exercise 4).

LIAPUNOV-SCHMIDT REDUCTION

176

Liapunov-Schmidt Reduction for O.D.E·s

In the following Sections we will increase the complexity of the Liapunov-Schmidt reduction program in steps.

While we dealt in the last

Section with bifurcations of steady-state solutions in one partial differential equation depending on one space variable. our next step will be to treat n coupled differential equations.

As a starting point we confine ourselves to n

coupled ordinary differential equations.

Since we consider only steady-state

solutions. we are effectively dealing with purely algebraic equations.

Hence a

Liapunov-Schmidt reduction for a steady state bifurcation in a system of o.d.e.·s is very simple.

The projections P and Q are simple products of two

~n_vectors involving the eigenvector corresponding to the eigenvalue zero.

Inverting the linear operator L amounts to inverting a matrix or solving a linear system of algebraic equations. Why would one want to perform a Liapunov-Schmidt reduction on ordinary differential equations when techniques are available (notably the center manifold reduction) which allow not only the determination of the bifurcated steady state solutions but also the dynamical behavior of the solutions approaching these steady states.

We offer three answers to this question:

First. it is instructive to see to what degree both methods give the same results and when and how they differ.

Second. there are criteria (see below)

which tell when the result of a Liapunov-Schmidt reduction can be augmented with some simple dynamics.

And third, a Liapunov-Schmidt reduction often

involves less computation than a center manifold reduction. In order to explain the latter fact. let us imagine a steady state bifurcation in an n-dimensional dynamical system (n say> 5).

If the

bifurcation is nondegenerate. then the center manifold is one-dimensional.

In

order to calculate the flow on the center manifold we perform nonlinear coordinate transformations on the whole n-dimensional system. although we are

LIAPUNOV-SCHMIDT REDUCTION

177

finally interested only in one Taylor coefficient, say the third order term for the flow on the center manifold.

Since the Liapunov-Schmidt reduction permits

us to find a specific coefficient in the bifurcation equation directly, less computation time is required for the Liapunov-Schmidt reduction than for the center manifold reduction.

This is supported by experience with both methods

for nontrivial research problems.

Problems where a MACSYMA program for a

center manifold reduction could not finish in a reasonable time were able to be handled by a Liapunov-Schmidt reduction program. To show the equivalence and differences of both methods we consider again the Lorenz system of Chapter 2.

x'

1

(35)

We calculated in Chapter 2 that for P

= lone

of the eigenvalues of (35) is

zero with the corresponding critical eigenvector [1,1,0]. L at p

= Pc = 1

associated with (35) is the matrix:

a

L

and L*

L

=

t

is the adjoint matrix.

[l/a,l,O].

The linear operator

[:

-1

[~

1

0

-~] 0

-1

0

0

-(3

This leads to the adjoint critical eigenfunction

The following is a sample run of a MACSYMA program REDUCTION2 to

LIAPUNOV-SCHMIDT REDUCTION

178

determine the bifurcation equation for this instability.

See Exercise 1 at the

end of the Chapter.

REDUcrION2() ; NUMBER OF EQUATIONS

3; ENTER Xl; ENTER X2; ENTER X3; ENTER RHO; ENTER 1;

VARIABLE NUMBER 1 VARIABLE NUMBER 2 VARIABLE NUMBER 3 THE BIFURCATION PARAMETER THE CRITICAL BIFURCATION VALUE RHO

WE DEFINE LAM = RHO - 1 ENTER THE CRITICAL EIGENVEcrOR AS A LIST [1.1.0]; ENTER THE ADJOINT CRITICAL EIGENVEcrOR [1/SIGMA.1.0]; ENTER THE DIFFERENTIAL EQUATION DIFF( Xl •T)= SIGMA*(X2-X1); DIFF( X2 •T)= Xl*X3+RHO*X1-X2; DIFF( X3 •T)= X1*X2-BETA*X3; [SIGMA (X2 - X1).- Xl X3 - X2 + (LAM + 1) Xl. Xl X2 - BETA X3] DO YOU KNOW A PRIORI THAT SOME TAYLOR COEFFICENTS ARE ZERO. YIN

N; TO WHICH ORDER DO YOU WANT TO CALCULATE 3;

2

2 d WI

2 d W3

d W2

2

[----- = O. ----- = O. ----- = ----J 2 dAMP

2

2

BETA

dAMP

dAMP

2 d WI

[--------- = dAMP dLAM

SIGMA

2 d W2

I

--------- = -------------------2 SIGMA

dAMP dLAM + 2 SIGMA + I

2 d W3

--------- = 0]

dAMP dLAM

2 SIGMA + 2 SIGMA + 1

LIAPUNOV-SCHMI1Jf REDUCTION [SYMBOLICS 3670 TIME

= 95

SEC.

179

J 3

AMP AMP LAM - ---BETA We find a bifurcation equation

3 gLorenz (A,h)= h A - (I/~) A

(36)

with nontrivial solutions

A

= ! (~

h)(1/2) which are identical to the

solutions found for the steady states on the center manifold (eq.(28) in Chapter 2).

In general these two reductions do not lead to identical results

for the steady state solutions but the topological character of the branching (i.e. here a pitchfork bifurcation) is the same in both cases.

However one can

* > 0 then the stability of the solutions of the show (cf.[I3J) that if (YI'YI> bifurcation equation g(A,h)

=0

with respect to perturbations in the direction

of the critical eigenfunction are given by the stability of the corresponding solutions of the dynamical system

dA

_ = g(A,h)

(37)

dt

Non-uniqueness of the Liapunov-Schmidt reduction and a change of stability is demonstrated by choosing a different critical eigenvector. We multiply the critical eigenvector by a term (-const) and perform the reduction:

REDUcrroN2() ;

some input tines omitted for brevity ENTER THE CRITICAL EIGENVEcrOR AS A LIST

[ -CONST. -CONST, OJ; some input tines omitted for brevity ...

[SIGMA (X2 - XI).- Xl X3 - X2 + (LAM + 1) Xl. Xl X2 - BETA X3J DO YOU KNOW A PRIORI THAT SOME TAYLOR COEFFICENTS ARE ZERO, YIN

N;

LIAPUNOV-SCHMIDT REDUCTION

180

TO WHICH ORDER DO YOU WANT TO CALCULATE 3'

2 d WI

2 d W2

[-----

2 d W3

2 2 (X)NST --------J BETA

0,

0,

2

2

dAMP

2 dAMP

dAMP

2

2

d WI

(X)NST

[---------

dAMP dLAM

SIGMA

d W2

2 SIGMA

illNST

dAMP dLAM

2

+ 2 SIGMA + 1

SIGMA

+ 2 SIGMA + 1

2 d W3

--------dAMP dLAM

= 72

[SYMBOLICS 3670 TIME

OJ SEC.J 3

AMP

3

(X)NST

- AMP (X)NST LAM + ----------BETA We find

=-

(38)

const A A + (const

3

/~)

3 A

o

as the bifurcation equation which has nontrivial solutions A

=~

( A pR) (1/2)

const Let us determine the stability of the trivial solution A dA

two reduced systems __ dt

gL

= x2 = x3 = 0

solution Xl

orenz

'

dA

00,

a + 1

2

of the original Lorenz system (35).

a

> -1

+ i

00,

-

~J

for the

= gL orenz and of the corresponding trivial

dt

eigenvalues of these systems become [_ a + 1 _ i

=0

00,

sign(const)

00,

For p

~

-

00

the

and

(cf. Chapter 2). respectively.

Hence for

2

and const

>0

we confirm that the solutions of dA dt

= gLorenz

have the

same stability as the corresponding solutions of the original system of o.d.e. 's. const

>0

* Also since the norm dA

then the solutions of __ dt

~

=1

+ I/o, we find that if

= gLorenz

0

< -1 and

have the same stability as the

corresponding solutions of the Lorenz system, thus verifying the aforementioned proposition of [13J.

LIAPUNOV-SCHMIDT REDUCTION

181

Application to Systems of Partial Differential Equations

Clearly there is no hope of writing a general program in MACSYMA which can deal with bifurcations in all physically interesting partial differential equations (p.d.e. 's) which may range over such fields as general relativity, plasmaphysics. hydrodynamics, elasticity and reaction diffusion equations.

The

major restriction to that task is not lack of memory space or slow performance (although this may become important), but rather the absence of suitable abstract datatypes in MACSYMA.

E.g. it is difficult to write a Frechet

derivative and it is not clear how to define function spaces like a Banach space, etc.

Hopefully the next generation of computer algebra systems (whose

first member is SCRATCHPAD II [46]) will give us the tools to work with differential operators on specifically defined function spaces. Therefore the program which we are going to describe now is designed to work for steady state bifurcations of a system of p.d.e. 's of the form

a at

where Lin[::2] is a linear differential operator of even order defined on some function space of

rn.

and NL is a nonlinear (differential) operator.

A typical

example of the type of equations which can be handled by this program are coupled reaction diffusion equations depending on one spatial dimension.

Since

the differential equation solver ODE2 in MACSYMA works only for one-dimensional o.d.e's of first or second order. we solve the differential equation for the Taylor coefficients of w(x;a,X), eq.(16.2). via a Fourier mode ansatz. this has its own problems:

However

Partial differential equations with Dirichlet

boundary conditions and quadratic nonlinearities lead to ordinary differential

LIAPUNOV-SCHMIDT REDUCTION

182

2

d w . equations for the Taylor coefficient ___ wh1ch have no finite Fourier mode 2 dA

expansion satisfying the boundary conditions (see Exercise 4 at the end of the Chapter.)

Instead, these o.d.e's display "resonant" terms.

Solutions to these

equations can readily be found for a single ordinary differential equation (see Exercise 4).

For n-coupled o.d.e. 's, this is not so easy and certainly goes

beyond the scope of this book.

Therefore the program we are giving here

cannot, in general, handle quadratic nonlinearities with sine-functions as critical eigenfunctions of the linearized problem.

(See however the specific

case in Exercise 5). The following example taken from Carr [6J shall demonstrate the program.

Let us consider the semi linear wave equation

(39)

with

V

v

=0

tt

+ v

at x

t

-

V

xx

= o,~.

- a v + fey)

0,

f(v)

v

3

5

+ p v •

Eq.(39) can be rewritten setting Y1

v

t

as

two coupled first order equations,

Y1

t

Y2

(40)

Y2

t

Yl

+ a Y

xx

1

3

- Y2 - Y 1

5

- P Y1

The linear operator L(a) is given by

(41)

where d

L(a)

xx

variable.

denotes the second partial derivative with respect to the spatial The eigenvalues of the linearized system are

LIAPUNOV-SCHMIDT REDUCTION

for eigenfunctions of the form [EF1 ] sin nx. EF 2

o Sa

183

Therefore for the interval

< 1, all eigenvalues have negative real parts and the trivial solution is

stable.

=1

For a

unstable.

+ is zero and the n = 1 mode becomes 1

the eigenvalue EV

The corresponding eigenfunction is CFUN

=[~ ]

sin x.

Setting

we find

u =

1r



f

u v + u (av + vI 2 I I 2

0

xx

- v 2 ) dx

1r

f

u v + u (av - v ) + u 2 2 1 2 I 2

0

vI dx

(integration by parts)

xx

1r

f

o

v (au + u l 2 2

xx

+ v (u -u ) dx 2 l 2

where we define the adjoint linear operator L* (a) by

[~ Hence the adjoint critical eigenfunction is given by y7 normalize such that
* 'Yl> =

1

I which gives us AFUN

= A sin

~

sin x

To calculate the bifurcation equation for (40) near a

now familiar ansatz Y =

t~]

substitute this into (40).

i ]. [ i ].

=

WI (X;A'A)] A sin(x) [] ~ + [w (x;A,A) with a 2

x [

We

I we make the

=A +

1 and

LIAPUNOV-SCHMIDT REDUCTION

184

Let us demonstrate how we can calculate the two Taylor coefficients

and

an example.

We show that the former coefficient is always zero:

2 d w

By

entering the ansatz into (40), differentiating twice with respect to A and setting A

= A = 0,

we obtain the two equations

o

A=O 1\=0

o

A=O 1\=0

which may be written in the form

[:~]

o

Now since L(l) y

=0

has only Y1 as nontrivial solution and since we require

that is zero, we have that the Taylor coefficient

2 d w

•

--- A=O 1S zero. dA2 1\=0 We could have concluded this right from the beginning, since (40) is invariant with respect to a reflection y

~

-y and hence the bifurcation equation and the

function W(X;A,A) inherit this symmetry and are equivariant with respect to the transformation A

~

-A [43J, i,e. all terms of even order in A are zero.

Our next step is to find the third order coefficient of w with respect to A.

If we set zw

1

=

3 d w 2

--A=O , we find after elementary dA3 A=O

calculations the following two equations for zw

1

and zw2 :

LIAPUNOV-SCHMIDT REDUCTION (42.1)

o

(42.2)

9/2 sin x - 3/2 sin 3x

185

Substituting (42.1) into (42.2) and using the boundary conditions zw X

= a,v,

we see that (42.2) has no solution.

1

=a

at

By the Fredholm alternative

theorem, since the homogeneous equation has a nontrivial solution, the nonhomogeneous equation will have a solution only if the nonhomogenity is orthogonal to the null space of the adjoint operator L*(1).

(cf. eq.(14.2»

This is guaranteed

by replacing [9/2 sin x 0 - 3/2 sin 3X] by

Then
is always zero.

v

2/v

f

o

We calculate

{9/2 sin2 x - 3/2 sin 3x sin x} dx

9/2.

Hence the Q-projected differential equations (42) become

- 9/2 sin x

9/2 sin x - 3/2 sin 3x

and a general solution to these equations with the zero boundary conditions at x

= a,v

is given by

LIAPUNOV-SCHMIDT REDUCTION

186 zW

= k

l

1

sin x + 3/16 sin 3x

-9/2 sin x

where k

is an undetermined constant.

1

Now since we require (Y1*'w> = O. we have to perform the same projection on this solution again. giving us. with

* [k 1 sin x + 3/16 sin 3X] > (Y1 ' - 9/2 sin x

-9/2 + k

1

the third order Taylor coefficient

3 d wI 3 dA 3 d w 2 dA

3

A=O

3/16 sin 3x + 9/2 sin x

1>=0

A=O

- 9/2 sin x

1>=0

Since the fourth order term is again zero for symmetry reasons, we can now calculate the Taylor coefficients of A for the bifurcation equation g(A.A) up to fifth order.

0

Evaluating the fifth order term is in principle the same as

for the one-dimensional bifurcation problems treated previously.

However the

projection P of eq.(14.1) now involves an inner product (13) which includes the scalar product with the adjoint eigenfunction y7 in ffi2 At this point we shall present a MACSYMA program, REDUCTION3. which accomplishes the Liapunov-Schmidt reduction for n coupled d.e.·s in one spatial variable.

We begin with a sample run on the bifurcation problem of eq.(40).

followed by the program listing:

LIAPUNOV-SCHMIDT REDUCTION REDUCTION3(); ENTER THE NUMBER OF DIFFERENTIAL EQUATIONS 2; ENTER THE DEPENDENT VARIABLES AS A LIST [Yl, Y2];

ENTER THE SPATIAL COORDINATE

X;

ENTER THE BIFURCATION PARAMETER ALPHA; ENTER THE CRITICAL BIFURCATION VALUE 1;

WE DEFINE LAM = ALPHA - 1 ENTER THE CRITICAL EIGENFUNCTION AS A LIST SIN(X)*[I,O]; ENTER THE ADJOINT CRITICAL EIGENFUNCTION AS A LIST 2/%PI*SIN(X)*[I,l]; ENTER THE DIFFERENTIAL EQUATION NUMBER 1

Y2; ENTER THE DIFFERENTIAL EQUATION NUMBER 2 DIFF(Yl,X,2)+ALPHA*YI-Y2-YlA3-A*YI A 5; 2 d Yl

5

[Y2, - Y2 + ---- - A Yl

3

- Yl

+ (LAM + 1) Yl]

2 dX

WHAT IS THE LENGTH OF THE SPACE INTERVAL %PI; DO YOU KNOW A PRIORI THAT SOME TAYLOR CXlEFFICIENTS ARE 0, YIN

Y; TO WHICH ORDER 00 YOU WANT TO CALCULATE 5; IS DIFF(W(AMP, 2 ,LAM, 0 ) IDENTICALLY ZERO. YIN

Y; IS DIFF(W(AMP, 3 ,LAM, 0 ) IDENTICALLY ZERO, YIN

N; 3 d WI

3 SIN(3 X) + 72 SIN(X)

3 d W2

9 SIN (X)

[----- =----------------------, ----- =- --------] 3

16

3

dAMP IS DIFF(W(AMP, 4 ,LAM, o Y; IS DIFF(W(AMP, 1 .LAM, 1 N; 2 d WI

[--------- = dAMP dLAM

2

dAMP

)

IDENTICALLY ZERO. YIN IDENTICALLY ZERO. YIN

2 d W2

SIN(X).

--------- = SIN(X)]

dAMP dLAM

IS DIFF(W(AMP. 2 ,LAM, 1 ) IDENTICALLY ZERO. YIN Y;

188

LIAPUNOV-SCHMIDT REDUCTION

IS DIFF(W(AMP. 3 .LAM. 1 ) IDENTICALLY ZERO. YIN N; 4

4

d WI

69 SIN(3 X)

+

2304 SIN (X)

d W2 18 SIN(X)]

[---------- = - -------------------------,

3

128

3

dAMP dLAM IS Y; IS Y; IS N; IS Y; IS N; IS N; IS Y; IS N; IS Y;

dAMP dLAM

G.YOLY( 1

o ) IDENTICALLY ZERO, YIN

G.YOLY( 2

0 )IDENTICALLY ZERO, YIN

G.YOLY( 3

0 )IDENTICALLY ZERO, YIN

G.YOLY( 4

o )IDENTICALLY ZERO. YIN

G.YOLY( 5

o )IDENTICALLY ZERO. YIN

G.YOLY( 1

1 )IDENTICALLY ZERO, YIN

G.YOLY( 2

1 ) IDENTICALLY ZERO, YIN

G.YOLY( 3

1 ) IDENTICALLY ZERO, YIN

G.YOLY( 4

1 ) IDENTICALLY ZERO, YIN

[SYMBOLICS 3670 TIME = 328 SEC.] 5

3

3 (- 1200 %PI A - 3195 %PI) AMP 3 AMP 3 AMP LAM + AMP LAM + ------------------------------ - -----1920 %PI 4 This is the bifurcation equation.

We manipulate this result to obtain A as a

function of A:

SOLVE(%.LAM); 4

2

(80 A + 213) AMP + 96 AMP [LAM = ---------------------------] 2

384 AMP + 128 TAYLOR(%,AMP ,0,4); 2

3 AMP

/T/

[LAM

+ ...

= -----4

4

(80 A - 75) AMP + ---------------- + . . . ]

128

LIAPUNOV-SCHMIDT REDUCTION

189

The following is a listing of the functions which perform the Liapunov-Schmidt reduction:

1* This file contains REDUCTION3(), a function to perform a Liapunov-Schmidt

reduction for steady state bifurcations from n coupled partial differential equations defined on one spatial dimension.

The following additional functions

are supplied: SETUP() allows the problem to be entered. G-POLY(I,J) calculates the coefficient of AMpAI LAMAJ in the bifurcation equation G(AMP,LAM). W-POLY(I,J) calculates the coefficient of AMpAI LAMAJ in W(X;AMP,LAM). SOLVE_ODE(EXP) solves certain ordinary differential equations via a Fourier mode ansatz. FEEDW(EXP) ensures that

=0

.

FIND_TRIG(EXP) identifies Fourier modes. SETIFY(LIST) transforms a list into a set. G-EQ() assembles the bifurcation equation. VFUN(LIST,VALUE) creates the substitution list: [LIST[1] = VALUE, LIST[2] = VALUE, ... ] DIFFNULL(I,J) sets the derivative diff(w,arnp,i,lam,j) to zero. *1

REDUCTION3():=BLOCK( SETUP(), ORDER:READ("TO WHICH ORDER DO YOU WANT TO CALCULATE"), FOR 1:2 THRU ORDER-1 DO W-POLY(I,O), FOR 1:1 THRU ORDER-2 DO

W-POLY(I,l) ,

FOR I: 1 THRU ORDER DO G-POLY(I, 0), FOR I: 1 THRU ORDER-1 DO G-POLY (1,1) , GJ:Q() )$

190

LIAPUNOV-SCHMIDT REDUCTION

SETUPC):=C /* This function performs the input for the Liapunov-Schmidt reduction*/ ASSUMEyOS:TRUE, LS-LIST: [] , N:READC"ENTER THE NUMBER OF DIFFERENTIAL EQUATIONS"), Y:READC"ENTER THE DEPENDENT VARIABLES AS A LIST"), XVAR:READC"ENTER THE SPATIAL COORDINATE"), ALPHA:READC"ENTER THE BIFURCATION PARAMETER"), CAL:READC"ENTER THE CRITICAL BIFURCATION VALUE"), PRINT C"WE DEFINE LAM = ", ALPHA - CAL), CFUN:READC"ENTER THE CRITICAL EIGENFUNCTION AS A LIST"), AFUN:READC"ENTER THE ADJOINT CRITICAL EIGENFUNCTION AS A LIST"), KILL(W) , W:MAKELISTCCONCATCW,I),I,1,N), ZWLIST:MAKELISTCCONCATCZW,I),I,1,N), NULLIST:MAKELISTCO,I,1,N), DEPENDSCAPPENDCZWLIST,W,Y),CONSCXVAR,[AMP,LAM])), EQS:MAKELISTCREADC"ENTER THE DIFFERENTIAL EQUATION NUMBER", I),!, 1 ,N), EQLAM:EVCEQS.EVCALPHA) = LAM + CAL), PRINT(EQLAM) • LEN:READC"WHAT IS THE LENGTH OF THE SPACE INTERVAL"), WNULL: VFUNCW, 0) , SUB:MAPLIST("=", Y,AMP*CFUN+W), DATABASE:APPENDCDIFNULLC1,0),DIFNULLCO.1)). ZWNULL: VFUN CZWLIST •0) • NORM: INTEGRATECAFUN. CFUN, XVAR. 0, LEN) , TEMPl:EVCEQLAM,SUB,DIFF), NULLANS:READC"OO YOU KNOW A PRIORI THAT SOME TAYLOR COEFFICIENTS ARE 0, Y/N")

)$

LIAPUNOV-SCHMIDT REDUCTION

191

G-POLY(I,J):=BLOCK( /*

This is a function to determine a particular Taylor coefficient of the

bifurcation equation G(AMP,LAM) = O. coefficients of W(AMP.LAM).

It requires kowledge about the Taylor

This knowledge is stored in the list DATABASE.

*/

LSJ.IST:OONS([K=I.L=J].LSJ.IST). IF NULLANS = Y THEN ( ZEROANS:READ("IS G-POLY(", 1,".". J, ") IDENTICALLY ZERO. Y/N"). IF ZEROANS = Y THEN RETURN(BIFEQ[I,J]:O). TEMP2:DIFF(TEMPl.AMP,I,LAM.J), DERIVSUBST:TRUE. /*

Set the derivatives diff(w,amp,i.lam,j) to zero. */

TEMP3:SUBST(DIFNULL(I.J),TEMP2). /*

Enter all information in DATABASE. */

D-BASEJ.ENGTH:LENGTH(DATABASE) , FOR I I THRU D-BASEJ.ENGTH DO TEMP3:EV(SUBST(DATABASE[D-BASEJ.ENGTH+I-II],TEMP3),DIFF), DERIVSUBST:FALSE, TEMP4:EXPAND(EV(TEMP3,AMP=O,LAM=O,WNULL,INTEGRATE»). /*

Project onto AFUN. */

TEMP5:RATSIMP(TEMP4.AFUN), BIFEQ[I,J):INTEGRATE(TRIGREDUCE(TEMP5).XVAR.O.LEN»)$

W-POLY(I,J):=BLOCK( /*

This function allows the iterative determination of any particular Taylor

coefficient of the function W(AMP,LAM).

It returns a differential equation for

the particular coefficient of interest (called ZWl.ZW2 ... ).

This d.e. is

solved via SOLVE_ODE and the result is fed into DATABASE from FEEDW. */ IF NULLANS = Y THEN ( ZEROANS:READ("IS DIFF(W(AMP,", 1," ,LAM,", J, ") IDENTICALLY ZERO, Y/N").

LIAPUNOV-SCHMIDT REDUCTION

IF ZEROANS = Y THEN ( ADDBASE:DIFNULL(I.J). DATABASE: APPEND(DATABASE ,ADDBASE), RETURN (ADDBASE) ) ) . WMAX-DIFF:MAP(LAMBDA([U]. 'DIFF(U.AMP,I.LAM,J) = CONCAT{Z,U)).W). TEMP2:DIFF(TEMPI.AMP.I,LAM.J), DERIVSUBST:TRUE. 1* Rename the derivatives diff(w.amp.i,lam,j) to zw *1

TEMP3:SUBST(WMAX-DIFF.TEMP2). 1* Enter all information stored in DATABASE *1

D-BASEJ...ENGTH: LENGTH(DATABASE) , FOR I I THRU D-BASL.LENGTH DO TEMP3:EV(SUBST(DATABASE[D-BASEJ..ENGTH+I-II].TEMP3).DIFF). DERIVSUBST:FALSE. TEMP4:EV(TEMP3.AMP=O,LAM=O. WNULL. INTEGRATE), 1* This is the projection Q onto the range of the

linear differential operator in the problem. *1 TEMP5:INTEGRATE(EV(TEMP4,ZWNULL).AFUN.XVAR.O,LEN), TEMP6: TEMP4-TEMP5INORM*CFUN, TEMP7:TRIGREDUCE(TEMP6). 1* The set of o.d.e.·s to solve.*1

W-DE:EXPAND(TEMP7). TEMP8:EV(W-DE,VFUN(ZWLIST,O)). 1* If the particular solution of W-DE is zero then w=O. *1

IF NULLIST=TEMP8 THEN ( ADDBASE:DIFNULL(I,J), DATABASE:APPEND(DATABASE.ADDBASE). RETURN(ADDBASE) ) • TEMP9:S0LVE_ODE(TEMP8). FEEDW(TEMP9)

)$

LIAPUNOV-SCHMIDT REDUCTION

SOLE_ODE(EXP):=(I* This function solves the d.e. W-DE by a Fourier mode ans:tz. *1 TRIGFUN: [] . CONST :FALSE. FOR I THRU N 00 ( 1* Determine the Fourier modes *1 TRIGl:EXP[I]. IF TRIGI # 0 THEN TRIG2:APPLYl(TRIGl.SINNULL.COSNULL), IF TRIG2 # 0 THEN ( CONST:TRUE, TRIGl:TRIGl-TRIG2). TRIGFUN:APPEND(FIND_TRIG(TRIGl),TRIGFUN))). SOLl:DELETE(DUMMY.SETIFY(TRIGFUN)). /* Make an ansa tz *1 ANSATZ:MAKELIST(SUM(AM[I.J]*SOLl[I],I.l.LENGTH(SOLl)),J.l,N), SOL2:EV(W-DE.MAPC"=" ,ZWLIST .ANSATZ) .DIFF). S0L3:MAKELIST(RATCOEF(SOL2,I),I.SOLl). EQLIST: []. FOR I THRU LENGTH(SOL3) 00 EQLIST:APPEND(EQLIST.SOL3[I]), VARLIST: []. FOR I THRU N 00 FOR J THRU LENGTH(SOLl) 00 VARLIST:CONS(AM[J.I].VARLIST). 1* Find the amplitudes of the Fourier modes *1 SOL4:S0LVE(EQLIST.VARLIST). /* Solve for the constant Fourier mode if necessary *1 CANSATZ:O, IF CONST = TRUE THEN (CANSATZ:MAKELIST(CONCAT(CON.I),I.l,N). (SOLl :EV(W-DE,MAP("=" .ZWLIST.CANSATZ) .DIFF).

LIAPUNOV-SCHMIDT'REDUCTION

194

CSOL2: APPLY1(CSOL1,SINNULL ,CXlSNULL) , CSOL3 :SOLVE(CSOL2, CANSATZ) ) , EV(ANSATZ+CANSATZ,SOL4,CSOL3»$

FEEDW(EXP):=(/* This function allows the result of the ode-solver to be entered into the list DATABASE.

It checks for orthogonality to the

critical adjoint eigenfunction and removes solutions of the homogeneous equation (i.e. nonorthogonal terms). */ F2: INTEGRATE(EXP. AFUN, XVAR, 0, LEN), IF RATSIMP(F2)=O THEN

ADDBASE:MAP("=",MAKELI'DIFF(W[U],AMP,I,LAM,J),U,I,N),EXP) ELSE (ORTHOYESULT: RATSIMP(EXP- F2/NORM*CFUN), ADDBASE:MAP("=",MAKELIST('DIFF(W[U],AMP,I,LAM,J),U,l,N), ORTHOYESULT)} , DATABASE: APPEND(DATABASE,ADDBASE) , PRINT(ADDBASE»$ /* Collect all information stored in BIFEQ and assemble the bifurcation equation. */ G-EQ():= SUM(EV(AMp K*LAM L/K!*BIFEQ[K,L],LS-LIST[N]),N,I,LENGTH(LS-LIST»$ A

A

SETIFY(LIST):=(/* Transforms a list into a set, i.e. removes duplicates. */ SET: [UST[l]], FOR I :2 THRU LENGTH(UST) DO IF NOT MEMBER(UST[I],SET) THEN SET:CXlNS(LIST[I],SET), SET)$

LIAPUNOV-SCHMIIJr REDUCTION

195

FINDJRIG(EXP):=(I* Finds the Fourier modes. */ LAI: ARGS(EXP+DUMMY). F....A2: APPLY1 (F.. ~U .SINFIND,(X)SFIND) )$

/* Auxiliary Functions */ MATaIDECLARE([DUMMYI.DUMMY2]. TRUE) $ DEFRULE(COOFIND,DUMMYIMCOS(DUMMY2) •COO(DUMMY2) )$ DEFRULE(SINFIND.DUMMYI*SIN(DUMMY2).SIN(DUMMY2»$ DEFRULE(COONULL,COO(DUMMY1).O)$ DEFRULE(SINNULL.SIN(DUMMY1).O)$

VFUN(LIST. VALUE): =MAP(LAMBDA([U], U=VALUE) ,LIST) $ DIFNULL(I.J):=MAP(LAMBDA([U],·DIFF(U,AMP,I.LAM.J)=D).W) $

Comments on the programs:

The main difference between these programs and those presented earlier in this Chapter is just bookkeeping: (i) Most of the variables which occur become n-dimensional lists.

E.g.the

unknown y is a N-dimensional list as well as the critical eigenfunction CFUN and its adjoint critical eigenfunction AFUN. N

(ii) The inner product involves a scalar product in IR

with subsequent

integration. (iii) In order to keep the amount of calculation to a minimum we also assumed again that the differential equations depend only linearly on the bifurcation parameter a.

This can, however, easily be changed by changing the iteration

range in the function REDUcrION3(). (iv) A major addition is the program called SOLVE-PDE which solves the

LIAPUNOV-SCHMIDT REDUCTION

196

differential equation associated with the Taylor coefficients of w(x;A.A) (cf. eq.(16.2).

It first identifies all trigonometric terms in a given expression

using the function FIND_TRIG and the rules COSFIND. SINFIND. COSNULL and SINNULL.

The function SETIFY removes multiple occurrences of the same

trigonometric functions such that one can make an ansatz for a solution of the differential equation W-DE. equation.

This ansatz is then inserted into the differential

We assume that these trigonometric functions are the solutions of

the linear eigenvalue problem for the original set of p.d.e.·s and form a complete set of orthogonal eigenfunctions on the defining space interval. Hence one can find the amplitudes of the ansatz by solving the algebraic equations which result from setting the coefficients of the trigonometric eigenfunctions to zero.
.

1S

The function FEEDW then guarantees that the projection

always zero.

(v) VFUN(LIST.VALUE) is a function which creates the substitution list [LIST[1J

= VALUE.

LIST[2J

= VALUE .... J

effectively.

Liapunov-Schmidt Reduction and Amplitude Equations

In this Section we shall use the Liapunov-Schmidt reduction to treat problems which involve coupled systems of nonlinear p.d.e.·s in more than one independent variable.

We begin with the description of an example.

We have dealt with the Lorenz system in Chapter 2 as well as earlier in this Chapter.

This system originated from an amplitude expansion and was

thought to describe the onset of two dimensional convection rolls [34J.

The

corresponding experimental situation is known as the 2D Rayleigh-Benard problem: above.

A fluid is confined to a box. heated from below and cooled from At a certain critical temperature (manifested in the Navier-Stokes

equations as a critical nondimensional number. the Rayleigh number Ra). the pure heat conductive state becomes unstable and stationary parallel convection

LIAPUNOV-SCHMIDT REDUCTION rolls develop.

197

This onset of convection is conveniently described by two

partial differential equations for the stream function

~(x,z)

(the

Navier-Stokes equations) and the deviation e(x,z) of the temperature from the purely conducting state, which are defined on a two-dimensional interval

o~

x

~

Lx' 0

~

z

Lz '

~

Usually one makes several approximations (the

Boussinesq approximation). of which the most important ones are incompressibility of the fluid, temperature independent material parameters and so-called free boundary conditions which allow for simple trigonometric solutions of the linearized problem.

The governing equations for the

stationary problem become:

Pr

-1

2

(dx~ d z - d z ~ dx ) v

~

(44) 2

- v

where v

2

= dxx

+ d

zz

e+

Ra

dx~

is the Laplace operator, and where Pr, the Prandtl number,

is a nondimensional number given by the quotient of the kinematic viscosity coefficient and the thermal diffusivity coefficient. Ra

It turns out that for

= 8, sin x sin z ] -4 cos x sin z

(45)

is a critical eigenfunction for the linearized problem (the left hand side of (44»

(46)

with eigenvalue zero.

With the adjoint linear operator defined by

-Ra d x 2 - v

we find the adjoint critical eigenfunction

LIAPUNOV-SCHMIDT REDUCTION

198

~:]

_ [

;in x sin z

e: - -2

(47)

cos x sin z

A bifurcation analysis shows that at Ra

= Ra c = 8

there exists a supercritical

pitchfork bifurcation to stable convecting rolls. described by the bifurcation equation

A A ~2/8 - A3 ~2/16

0 with, as usual, A

= Ra

- RaC'

Supercritical

means that if one varies A and approaches the bifurcation point on the branch A

=0

from the stable side, then the branching solutions bifurcate in the

forward direction (cf. Fig.4 in Chapter 1).

Using the o.d.e. (37) it can

easily be shown that supercritical solutions are stable and subcritical solutions (those branching backwards) are unstable. In the following sample run of the Liapunov-Schmidt reduction for this problem we show how we can use slightly extended versions of the building blocks SETUP, G-POLY, W-POLY and FEEDW given in the previous section to deal with coupled partial differential equations in more than one space variable. Since it is impractical to write a general program which performs a Fourier transformation of the coupled partial differential equations which have to be solved to find w(a;A,A), we perform this step interactively.

The solution is

then given as input to the function FEEDW which puts the result into DATABASE.

LAPLACE(FUN) := DIFF(FUN.X,2)+DIFF(FUN,Z,2)$ Warning - you are redefining the MACSYMA function LAPLACE SETUP() ; ENTER THE NUMBER OF DIFFERENTIAL EQUATIONS 2; ENTER THE DEPENDENT VARIABLES AS A LIST [PSI, THETA]; ENTER NUMBER OF SPATIAL aXlRDINATES 2;

ENTER THE SPATIAL aXlRDINATES AS A LIST [X.Z] ; ENTER THE BIFURCATION PARAMETER RA; ENTER THE CRITICAL BIFURCATION VALUE 8; WE DEFINE LAM = RA - 8 ENTER THE CRITICAL EIGENFUNCTION AS A LIST [SIN(X)*SIN(Z),-4*OOS(X)*SIN(Z)];

LIAPUNOV-SCHMIDT REDUCTION

199

ENTER THE ADJOINT CRITICAL EIGENFUNCTION AS A LIST [SIN(X}*SIN(Z},-COS(X}*SIN(Z)/2]; ENTER THE DIFFERENTIAL EQUATION NUMBER 1 (DIFF(PSI,X)*DIFF(LAPLACE(PSI),Z)-DIFF(PSI,Z)*DIFF(LAPLACE(PSI),X»* PRANDTL~-l-(LAPLACE(LAPLACE(PSI»-DIFF(THETA,X»;

ENTER THE DIFFERENTIAL EQUATION NUMBER 2 DIFF(PSI,X)*DIFF(THETA,Z)-DIFF(PSI,Z)*DIFF(THETA,X)-RA*DIFF(PSI,X} +LAPLACE(THETA); ENTER THE LOWER LEFT CORNER OF THE 2 DIMENSIONAL SPACE INTERVAL [X

= ... ,] 1

[X

= O,Z = 0];

ENTER THE UPPER RIGHT CORNER OF THE 2 DIMENSIONAL SPACE INTERVAL

=... ,]

[X 1

[X

= %PI,Z = %PIJ;

= 270

[SYMBOLICS 3670 TIME

SEC.]

dPSI

3 d PSI

3 d PSI

3 d PSI

3 d PSI

dPSI

(----- + ------) - (----- + ------)

dX

4

dTHETA

3

2

3

2dZ

d PSI

dZ dX dZ dX dX dZ [------ - ----- + --------------------------------------------dX 4 PRANDTL

dZ 4 4 d PSI d PSI - ----- - 2 -------, 4

2

dX

dX

2 d THETA

2

dZ dPSI dTHETA

2 d THETA

dPSI dTHETA

------- + ---- ------ + ------- - ---- ------ -

2

dX

dZ

dZ

2 dX

dZ

(LAM

+

dX

dPSI 8) ----] dX

GJ'OLY(l,O) ; [SYMBOLICS 3670 TIME = 13 SEC. ]

v This is a check on whether we solved our linear problem correctly. dealing with a bifurcation problem, this coefficient is always zero.

WJ'OLY(O, 1);

If we are

LIAPUNOV-SCHMIDT REDUCTION

200 NOW SOLVE THE EQUATIONS 4 dZW2 d ZW1

4 d ZW1

2

4 d ZW1

2

d ZW2 d ZW2 dZW1 [---- - ----- - ----- - 2 -------. ----- + ----- - 8 ----J dX 4 4 2 2 2 2 dX dZ dX dX dZ dZ dX ::O! THEY ARE GIVEN IN W-DE

[SYMBOLICS 3670 TIME

=5

SEC.J CALL FEEDW() TO PROCEED

This is the Linearized equation. and so the onLy nontriviaL soLution is given

by CFUN.

Since

= O.

we find that ZWI

= ZW2 = O.

h:rnw(); d;

ENTER ZW1 ENTER ZW2 0:

[SYMBOLICS 3670 TIME

=3

SEC. J dW1

dW2 ----

dLAM

dLAM

[---- = O.

= OJ

WPOLY(2.0);

4 4 4 dZW2 d ZW1 d ZW1 d ZW1 [---- - ----- - ----- - 2 -------. dX 4 4 2 2 dZ dX dX dZ

NOW SOLVE THE EQUATIONS

2

2

d ZW2 d ZW2 dZW1 2 ----- + ----- - 8 ---- + (- 4 SIN (xj

2 dZ

2 dX

2

- 4 COS (X»

SIN(2 Z)J

dX ::O! THEY ARE GIVEN IN W-DE

tSYMBOLICS 3670 TIME

= 15

SEC. J CALL FEEDW () TO PROCEED

TRlGREDUCE(W-DE.X) ; [SYMBOLICS 3670 TIME = 1 SEC.] 4 d ZW1

4 d ZW1

4 d ZWI

2 d ZW2

2

d ZW2 dZWl [---- - ----- - ----- - 2 -------. ----- + ----- - 8 ---- - 4 SIN(2 Z)J 2 2 dX dX 4 4 2 2 dZ dX dX dZ dZ dX

dZW2

LIAPUNOV-SCHMIDT REDUCTION

201

EV(%.ZWI = AMPl*SIN(2*Z).ZW2 = AMP2*SIN(2*Z).DIFF); [SYMBOLICS 3670 TIME = 1 SEC.] [- 16 AMPI SIN(2 Z). - 4 AMP2 SIN(2 Z) - 4 SIN(2 Z)] FEEDW(); ENTER ZWI O' ENTER ZW2 -SIN(2*Z) ; [SYMBOLICS 3670 TIME

13 SEC.] 2 d

2

WI

d

[----- = O.

W2 - SIN(2 Z)]

2 dAMP

2 dAMP

BIFEQ:G-POLY(I.I)*AMP*LAM+G-POLY(3.0)*AMp 3/3!; A

[SYMBOLICS 3670 TIME = 1 SEC.] 2

2

%PI AMP LAM 8

%PI

3 AMP

16

Since the program largely duplicates the corresponding functions given in the the previous Section. we do not supply a complete listing of the functions used here. but rather we only discuss some important new features. The major difference between the programs used here and those given in the previous Section is that we are dealing with more than one independent variable. say SPACE variables.

These have to be entered into a list. say XVAR.

and the dependency of the independent variables Y and the variables W and ZW on XVAR has to be declared.

Also the projections P and Q now involve integration

over several space intervals.

This is most effectively performed via the

following auxiliary function.

(Do not use multiple definite integration in

MACSYMA.

It is very slow and keeps asking you annoying questions about the

sign of totally uninteresting quantities):

LIAPUNOV-SCHMIIJf REDUCTION

202 INT(EXP) :=(INTINT: EXP ,

FOR I THRU SPACE DO INTINT:INTEGRATE(TRIGREDUCE(INTINT.XVAR[I]).XVAR[I]). RATSIMP(EV(INTINT,UBOUND) - EV(INTINT.LBOUND»$

where LBOUND is the lower left corner of the SPACE-dimensional simplex given as .~=

a list [X = I

•... ] and UBOUND is the corresponding upper right corner.

What has all this to do with the Lorenz equations?

We observe from the

last demonstration run that for the steady state bifurcation of the Navier Stokes equations there are only three relevant modes for a bifurcation equation up to third order: the two modes of the critical eigenfunction sin x sin z and cos x sin z and the perturbation mode in the temperature equation sin 2z. one makes the ansatz

[~.e] = [ A

v8

sin x sin z·. -4

v8 B

cos x sin z

- 8 C sin 2z ]

we can recover the Lorenz model, as we show next:

EQS;

2 dTHETA

2 d PSI

d

[------

(----- + -----)

2

dX dZ

dPSI dX

2 d PSI

2

2 d PSI

d

2

dZ

dX

2 d PSI

(-- (----- + dZ 2 dZ

dPSI

-----»

dZ

2 dX

2 2 d d PSI d PSI (-- (----- + -----» dX

2 dZ

2 dX

+ -----------------------------------------------------

PRANDTL

2

2 d

2

2

d

2

d PSI (----~

2 dX

2 dZ

dT

d PSI + -----)

2 dX

d

PSI

d PSI

(----- + -----) 2 2 dZ

dX

+

PRANDTL

If

LIAPUNOV-SCHMIDT REDUCTION 2

203

2

d THETA

dPSI dTHETA

d THETA

dPSI dTHETA

THETA

------- + ---- ------ + ------- - ---- ------ - ------ -

2 dZ

dX

dZ

2 dX

dZ

dX

dT

dPSI RA ----] dX

EV(EQS,PSI = A*2*2A(1/2)*CFUN[1], THETA = B*2*2A(1/2)*CFUN[2]~8*SIN(2*Z).DIFF)$ TRIGREDUCE(%.X)$ TRIGREDUCE(%, Z); dA

4 SQRT(2) -- SIN (X) SIN(Z) dT [- -------------------------- +

PRANDTL

8 SQRT(2) B SIN(X) SIN(Z)

- 8 SQRT(2) A SIN(X) SIN(Z). SIN(3 Z) SIN(Z) 2 SQRT(2) A (- 16 C OO8(X) (-------- - ------) 2

2

- 2 SQRT(2) B 008(2 X) SIN(2 Z) - 2 SQRT(2) B SIN(2 Z» 1 008(2 X) dC - 16 A B (- - --------) SIN(2 Z) + 8 -- SIN(2 Z) + 32 C SIN(2 Z) 2 2 dT dB

- 2 SQRT(2) A RA OO8(X) SIN(Z)

+

8 SQRT(2) -- OO8(X) SIN(Z) dT

+ 16 SQRT(2) B OO8(X) SIN(Z)]

SOLVE([RATCOEF(%,OO8(X)*SIN(Z»[2].RATCOEF(%,SIN(X)*SIN(Z»[1], RATCOEF(%,SIN(2*Z»[2]],[DIFF(A,T),DIFF(B.T).DIFF(C,T)]); dB

dA

[[-- = 2

B PRANDTL - 2 A PRANDTL,

ARA-8AC-8B

-- = -------------------

dT

dT

4

dC

-- = 2

A B - 4 C]]

dT

Note that with rescaling time d/dt

~

2( )' and setting

~

= 2.

RA

= 8p.

the amplitude equations for A.B.C are identical to the Lorenz equations (35). Even the bifurcation equation for the Liapunov-Schmidt reduction from the Navier-Stokes equations and the reduction starting with the Lorenz model are identical for p

= 2.

LIAPUNOV-SCHMIDT REDUCTION

204

Whereas most of the amplitude expansions which are popular especially in hydrodynamics seem to be constructed on physical arguments, we can offer here a rigorous procedure to determine the necessary modes for an ansatz (cf.[15]):

If we want to be exact up to a certain order, we have to take into

account all the modes which occur in the Liapunov-Schmidt reduction up to this order.

Note that this does not require that we actually perform the reduction.

It is much simpler to find the relevant modes than to perform the reduction, especially since it does not involve solving the differential equation (16.2) but only orthogonal expansions as in the foregoing sample run (cf. also the following Section).

Hence sometimes it pays to do a "reduced" Liapunov-Schmidt

reduction, i.e., determining only the relevant modes and then making an amplitude expansion with respect to these modes.

Then one can use such

powerful techniques as normal form transformations or center manifold reductions on the set of modal equations.

These are o.d.e.'s and hence involve

only polynomial manipulations for which MACSYMA is much better equipped than for dealing with p.d.e.'s.

A last remark: Obviously these modal expansions

cannot say anything more about the true solutions of the p.d.e. than the corresponding Liapunov-Schmidt reduction.

In particular all solutions are

valid only locally and are isomorphic to the true solutions up to the degree for which the modes of the amplitude expansion correspond to the modes in the Liapunov-Schmidt reduction.

Hence the chaotic solutions for which the Lorenz

system is famous are not directly related to the solutions of the Navier-Stokes equations.

LIAPUNOV-SCHMIDT REDUCTION

205

Determination of Higher Order Coefficients

If we try to find the bifurcation equation for some nontrivial partial differential equations up to higher order we may run into trouble.

For

instance the intermediate expressions which one encounters using the function G-PQLY and W-PQLY on the.Benard problem to determine a fifth order coefficient are quite large.

This situation becomes even worse when we think of

calculating bifurcation equations up to higher order for degenerate problems. i.e. bifurcations with more than one critical eigenfunction becoming simultaneously unstable.

In principle the procedure is straightforward: For a

problem with n critical eigenfunctions we have an n-dimensional kernel N which is spanned by Alu c . A u c .' .. Anu nc • the n critical eigenfunctions and their l 2 2 amplitudes.

The projection P onto the kernel leads now to n coupled algebraic

bifurcation equations gl(A 1 ....• An.A) ....• gn(A1 ....• An.A) and the function w which lies in the complement of N is also a function of n amplitudes w(x;A ..... An.A). 1

Hence it is conceptually not difficult to extend the

function G-POLY and W-POLY to the n-fold degenerate steady state bifurcation problem.

However. due to intermediate expression swell. this program will be

of very limited use for coefficients higher than third order.

Also we

encounter one of the basic issues of Computer Algebra: Even if it is possible to calculate nontrivial fifth order coefficients. these are usually large and complicated algebraic expressions. depending on the parameters of the problem. In general these expressions are too difficult to interpret and hence are meaningless,

However in'some interesting cases we have been able to reduce the

number of coefficients involved by fixing them at critical values such that lower order terms in the bifurcation equations vanish.

Then the next higher

order terms are the leading order terms and therefore their signs determine the whole local bifurcation structure.

If we are lucky then the relevant

coefficients depend only on one or two parameters and can be plotted as

LIAPUNOV-SCHMIDT REDUCTION

206

functions of these parameters. The following interactive procedure describes a highly efficient way of calculating these higher order coefficients and plotting their graphs.

Its key

requirement is that products of the critical eigenfunctions must have a finite decomposition into a complete set of suitable orthogonal eigenfunctions.

For

trigonometric eigenfunctions this means that we need such nonlinear terms that the Fourier decomposition of the occurring products of the critical eigenfunctions are finite.

The aim of the procedure is to create a FORTRAN

program out of MACSYMA which eventually feeds into a plotting routine. Rather than presenting the procedure in full generality, we treat the Benard problem of (44) as an example (cf.[1],[21]).

We write the Boussinesq

equations as

(48)

NL (..p.B)

where ..p.B are the stream function and the temperature variation respectively, L is the linear operator and NL (..p,B) a vector-valued nonlinear differential operator.

For a 2-dimensional infinitely extended problem and "free"

horizontal boundary conditions at z

= -1/2

and 1/2, we find two simultaneously

unstable critical modes at a critical periodicity length defined by

..pc

= [A

sin(ax+~)

+ B sin(2ax+x)] sin

a:

v(z+~)

(49)

B c

(50)

2 2 2 -1 - [A (v +a) a

cos(ax+~)

2 2 2 -1 . 1 + B (v +4a) (2a) cos(2ax+X)] SIn V(Z+2)

LIAPUNOV-SCHMIDT REDUCTION For this degeneracy the critical Rayleigh number Ra wavelength a are given by

Ra

c

= 769.234,

a

207 and the critical

c

= 1.5501534.

a very large Benard problem (length of container

»

In this way we model

l/a) as an infinitely

extended problem with periodic boundary conditions.

The critical wavevector a

then corresponds to a periodicity length L such that the modes a and 2a become simul taneously unstable .. Such an L can always be found.

Since we impose only

periodic boundary conditions we allow for arbitrary variations in the phases

~

and )(. Symmetry arguments [1] reveal that the generic system for this problem is of the form

A'

= ( A + all

A2 + a 12 B2) A + c l A3B2 cos

2~

(51)

- A2 (2c 1 B2 + c 2 A2 ) sin where

~

= 2~

-)(.

2~

The first occurrence of the phase

equations is in fifth order terms.

~

in the bifurcation

These terms are crucial for the solution

structure of the bifurcation equations (51).

To determine them we make the

following ansatz:

(52)

where wt .. we. 1

1

= O(A i+1 ,B i+l ).

Clearly in order to determine a fifth order term

in the bifurcation equation for a p.d.e. with quadratic nonlinearity we have to ~ow

the function w, i.e. here wt and we, to order four in A and B.

The key

LIAPUNOV-SCHMIDT REDUCTION

208

observation to reducing the necessary calculations is that, in order to determine a specific fifth order coefficient, we do not need w up to fifth order exactly.

We only need to know certain Fourier coefficients of it.

We proceed stepwise: Step 1:

Insert (52) into (48) and expanq to order 2 in A and B.

This

gives:

L [

(53)

since L

[~c

W'#1 ] w8 1

= NL

] = 0 by definition.

(~

c

,8 ) c

Now we use the command TRIGREDUCE in

c

pn the right hand side of equation (53) to give NL

(~

~CSYMA

,8 ) decomposed into its

c

c

fourier modes, say

f1trig

w'lJ~

Here trig w'#t and trig W8~ with 1 ~

j

~ n, 1 ~ I ~ m stands for one of the n

and m possible Fourier modes in w'#k and w8 (cf. eq.(52)), respectively. k

E.g.

we find for the Boussinesq eq.(48)

The coefficients f

i

and g. depend on A or B or both. 1

Now we set

w'#l

. w'lJl1 = c 11 tng

+ ...

c

n ~n . 1 tng w 1

w8

= d 1I trig w8~

+ ...

m . wam1 d tng 1

(54) 1

where the coefficients c

i

1

dI are as yet unknown functions of 0(2) in A and B.

LIAPUNOV-SCHMIDT REDUCTION

209

which have the same argument as the corresponding coefficients fi,gj'

We do

not solve (53) in this step. Step 2:

Insert (52) with

to order 3 in A and B.

(~I,w91)

given by (54) into (48) and expand

This gives:

where NL is that part of the Boussinesq equations which is quadratic in (cf.(44».

~

We convert the right hand side again to Fourier modes whose

1 n 1 m coefficients now depend on A,B,c ,.· .,c ,d , ... ,d and set l l l l

.1,

__

w~

C21

trIg '

.1,1 + . . . + Cv WP2 2

V trIg ' "". .P2

(56)

where v and s are the number of Fourier modes in

~2

and w9

2

respectively.

Again, the c;, d4 have the same functional dependence on the above set of parameters as the corresponding coefficients of the r.h.s. of (55).

We proceed

in this way to order 5 and project the nonlinear part of 0(5) onto the eigenspaces given by

and

In each of the resulting

algebraic equations there is only one term which depends on the phases namely cos

2~-X,

~

and X'

which are the coefficients c

and c of (51), respectively. 2 l j ffi . i d k . turn . They d epend on t h e prevIous coe Clents c , 3' c ' wh'IC h In 3 2 determine the crucial modes trig fifth order terms c

modes.

In order to evaluate c

relevant for the

and c ' we trace back the 2 l above calculation and determine in each step the inverse L- for these critical l

and c . 2

w~;, trig w9~, trig w~~,

In this way we find the coefficients j

1

coefficients c - and d i _ l i l

l

c~I as functions of the

However we solve each step separately such that

LIAPUNOV-SCHMIDT REDUCTION

210

we never accumulate large algebraic expressions.

Instead we write the results

out from MACSYMA into FORTRAN and finally plot a graph c c

2

= c2 (Pr).

1

= c 1 (Pr),

Our example shows the efficiency of this method: There are of the

order of 100 modes in the function w which contribute to fifth order. for the coefficients c

1

and c

W'/J 1 "" sin(ax+J(-.p) sin

2

the only relevant modes are:

2Jr(Z+~)

W'/J2 "" [sin(J(-2.p) + sin(3ax+2J(-.p)] sin

v(Z+~)

+ [sin(J(-2.p) + sin(3ax+2J(-.p)] sin 3v(z+~)

W'/J 3 "" [sin(ax-J(+3.p) + sin(2ax+2J(-2.p)] sin 2v(z+~)

and corresponding cosine modes for wei' i

1.2.3.

However

LIAPUNOV-SCHMIDT REDUCTION

211

Exercises

1.

Change the program REDUCTIONI such that it runs correctly and most

efficiently for steady state bifurcations from o.d.e. 'so

Reproduce the results

of the sample run of REDUCTION2 on the Lorenz equations (35).

Hint: You are dealing with purely algebraic equations.

Here the differential

equation (16.2) reduces to a system of nonlinear algebraic equations.

The

resulting equations for the Taylor coefficients of W(X;A,A) are systems of linear equations which can easily be solved automatically.

2.

A variant of the problem of eqs.(39}.(40} is the following

integrodifferential equation [6] which describes the motion of an elastic rod with hinged ends.

(PI)

wi th Yl

= Y2 = ° at

x

= 0,1.

Use the program REDUCTION3 to show that the

bifurcation equation up to fifth order for (PI) is given by

2 2 3 3 5 -11" M-411" M -A -3A =0

LIAPUNOV-SCHMIDT REDUCIION

212 3.

The semilinear wave equation (39) can be treated in an alternate fashion:

Since we are only interested in stationary bifurcations, we can perform the Liapunov-Schmidt reduction on the one-dimensional differential equation

(P2)

with v

-

=0

at x

3

- av + V

V

xx

= O.n.

+ pv

5

o

Use the program REDUCTION1 to determine the bifurcation

equation up to fifth order for (P2).

Compare your result with the result

obtained in the text by using REDUCTION3. The differences can be understood in the following way:

two dimensional version is proportional to [ the one dimensional version

3 d w -A=O for the 3 dA A=O

c sin x + d sin 3X] _ c sin x ' whereas for

is proportional only to sin 3x.

Hence the

fifth order coefficient in the bifurcation equation in the former case contains an additional term of the form

Sn o

2 sin x p sin x

This is a good example of the nonuniqueness of the Liapunov-Schmidt reduction.

Note however that the local information for A,A near zero is the

same in both cases.

This is manifested by the solutions of the bifurcation

equation which are A

= 3/4

2 A + terms of higher order, where the two reductions

differ only by these higher order terms.

For a more extensive discussion of

the equivalence of different Liapunov-Schmidt reductions and the question where to truncate a bifurcation equation, see [13].

LIAPUNOV-SCHMIDT REDUCTION

213

4. Calculate the bifurcation equation for the pne-dimensional problem

y"

(P3)

+ a y + y2

with Dirichlet boundary conditions yeO) For a

2

=V

the form sin vx.

o

= y(l) = O.

the trivial solution is unstable against a perturbation of Note that the differential equation for

d

2

2

w

dA

A=O

( = zw)

A=O

becomes

(P4)

zw

+ v

2

zw

2 - 2 sin vx

- (1 - cos 2vx).

If we try to solve this equation via a Fourier transformation with those Fourier modes that satisfy the boundary conditions (i.e. a pure sine-series), we find that the solution

Zw

is given by an infinite series.

The approach in

REDUCTION1 avoids this by solving the o.d.e.(P4) directly, introducing a "resonant" term of the form x sin vx.

does

Check that the solution for

indeed satisfy the boundary conditions and is orthogonal to sin vx.

Can you

think of a way of extending this approach to n-dimensional systems, n>l?

LIAPUNOV-SCHMIDT REDUCTION

214

5. The Brusselator is a well-known example of a reaction diffusion equation [33].

Its steady state bifurcations from a homogeneous distribution of two

chemical substances Yl and Y2 is described by

(P5)

where

and where a and b are externally controllable chemical substances and d and d

2

d

xx

parameter.

describe the diffusion.

d

1

xx

Usually one treats b as the bifurcation

Let us define the system on the interval [O,v] with Dirichlet

boundary conditions. Show that for a 2 = d d , (P5) has a zero eigenvalue at b 1 2

= (1

+ d )2

1

with critical eigenfunction sin x [ _ ;2_ d ] and adjoint critical 1 eigenfunction sin x [ 1 +11/d1 ]. order.

Find the bifurcation equation up to second

Determine why the program SOLVE-PDE encounters inconsistent equations

when attempting to calculate W-POLY(2.0) (cf. exercise 4 and [4]). for d

1

=1

Show that

the first nonvanishing term in the bifurcation equation is of third

order and calculate this term.

APPENDIX

INTRODUCfION TO MACSYMA

Introduction

The purpose of this Appendix is to provide an introduction to the computer algebra system MACSYMA.

We have included this introduction so that

the book would be understandable to the reader with no familiarity with

MACSYMA.

However, it is by necessity so brief that the reader is referred to

other sources for further information ([26].[35]). If, as we intend, the reader plans to use our MACSYMA programs to perform perturbation calculations. it is unquestionably desirable that some first-hand experience with MACSYMA be available before trying our programs. For a reader with no MACSYMA experience, we suggest reading this introduction in front of a computer terminal running MACSYMA.

Then the reader may wish to

duplicate our run. punctuated with explorations of the reader's design. end of the Appendix we offer some elementary exercises for the beginning

MACSYMA user.

To invoke MACSYMA on most machines, type:

MACSYMA (return)

The computer will display a greeting of the sort:

At the

INTRODUCTION TO MACSYMA

216

This is MACSYMA 304 [copyright message ...

J

Loading fix file <MACSYM>TOPS20.FIX.304

(CI)

The (CI) is a "label".

Each input or output line is labelled and can be

referred to by its label for the rest of the session.

C labels denote your

Commands and D labels denote Displays of the machine's response.

Never use

variable names like CI or D5, as these will be confused with the lines so labeled.

Special Keys and Symbols

1. To end a MACSYMA session, type QUIT(); . will also work.

(Here

A

On some machines AC or Ay or AZ

stands for the control key, so that

~

means

simultaneously press both the key marked control and the C key). 2. To abort a computation without leaving MACSYMA, type AG or AC or press the ABORT key.

This step will differ from machine to machine.

It is important for

you to know how to do this on your machine, in case, for example,

you begin a

computation which is taking too long. 3. In order to tell MACSYMA that you have finished your command, use the semicolon (;).

On some machines you must follow the; with a return.

Note

that the return key alone does not signal that you are done with your input. 4. An alternative input terminator to the semicolon (;) is the dollar sign ($), which, however, supresses the display of MACSYMA's computation.

This is 4seful

if you are computing some long intermediate result, and you don't want to waste time having it displayed on the screen.

INTRODUCTION TO MACSYMA

217

5. If you want to completely delete the current input line (and start this line fresh from the beginning), type a double question mark (??). 6. If you wish to repeat a command which you have already given, say on line (05), you may do so without typing it over again by preceding its label with a double quote ("), i.e., "05. (Note that simply inputing 05 will not do the job - try it.) 7. If you want to refer to the immediately preceding result computed by MACSYMA, you can either use its D label, or you can use the special symbol percent (%). 8. The standard quantities e (natural log base), i (square root of -1) and pi (3.14159 ... ) are respectively referred to as %E, %I and %PI.

Note that the use

of % here as a prefix is completely unrelated to the use of % to refer to the preceding result computed. 9. In order to assign a value to a variable, MACSYMA uses the colon (:), not the equal sign.

The equal sign is used for representing equations.

Arithmetic

The common arithmetic operations are: + addition

- subtraction

*

scalar multiplication

/ division A

or ** exponentiation matrix multiplication factorial

SQRT(X) square root of X MACSYMA's output is characterized by exact (rational) arithmetic.

E.g.,

218

INTRODUCTION TO MACSYMA

(C1) 1/100+1/101; 201 (Dl)

10100 If irrational numbers are involved in a computation, they are kept in symbolic

form:

(C2) (1+SQRT(2»)A5; 5

(SQRT(2) + 1)

(D2) (C3) EXPAND(%);

29 SQRT(2) + 41

(D3)

However, it is often useful to express a result in decimal notation.

This may

be accomplished by following the expression you want expanded by ",NUMER":

(C4) %,NUMER; 82.012194

(D4)

Note the use here of % to refer to the previous result.

In this version of

MACSYMA, NUMER only gives 8 significant figures, of which the last is often unreliable.

However, MACSYMA can offer arbitrarily high precision by using the

BFLOAT function:

(CS) BFLOAT(D3);

(D5)

8.201219330881976B1

The number of significant figures displayed is controlled by the MACSYMA variable FPPREC, which has the default value of 16:

(C6) FPPREC; (00)

16

Here we reset FPPREC to yield 100 digits:

(C7) FPPREC:100; (D7)

100

(CS) "CS;

(DS) 8.20121933088197564152489730020812442785204843859314941221237124017312# 4187540110412666123849550160561B1

INTRODUCTION TO MACSYMA

219

Note the use of the double quote (' ') in (OS) to repeat command (C5).

MACSYMA

can handle very large numbers without approximation:

(C9) 100!; (09) 9332621544394415268169923885626670049071596826438162146859296389521759# 999322991560894146397615651828625369792082722375825118521091686400000000000# 0000000000000

Algebra

MACSYMA's importance as a computer tool to facilitate perturbation calculations becomes evident when we see how easily it does algebra for us.

Here's an

example in which a polynomial is expanded:

2

3

(X Y + 3 Y + X)

(Dl) (C2) EXPAND(%);

6343 23 352 (D2) X Y + 9 X Y + 27 X Y + 27 Y + 3 X Y 3

+ 18 X

2 2 4 2 3 Y + 27 X Y + 3 X Y + 9 X Y + X

Now suppose we wanted to substitute 5/Z for X in the above expression:

(C3) D2,X=5/Z; 2

3

135 Y

675 Y

2

225 Y

3

2250 Y

125

5625 Y

(D3) ------ + ------ + ----- + ------- + --- + 2 2 3 3 Z Z

Z

Z

4 Z

Z 2

3

1875 Y 9375 Y 15625 Y 3 + ------ + ------- + -------- + 27 Y 4

Z

5

Z

6

Z

The MACSYMA function RATSIMP will place this over a common denominator:

INTRODUCTION TO MACSYMA

220 (C4) RATSIMP(%); 3

6

(D4) (27 Y 2

2

5

+ 135 Y 2

3

4

+ (675 Y + 225 Y) 2

2 3 3 2 2 + (2250 Y + 125) 2 + (5625 Y + 1875 Y) 2 + 9375 Y 2 3 6 + 15625 Y )/2

Expressions may also be FACTORed:

(CS) FACTOR(%); 2

(3 Y 2

3

+ 5 2 + 25 Y)

(DS) 6

2

MACSYMA can obtain exact solutions to systems of nonlinear algebraic equations. In this example we SOLVE three equations in the three unknowns A,B,C:

(C6) A+B*C=I; (00) (C7) B-A*C=O;

BC+ A= 1

(m)

B-AC=O

(CS) A+B=5; (DS)

B+ A= 5

(C9) SOLVE([D6.D7,D8],[A,B,C]); 25 SQRT(79) %1 + 25

(D9) [[A

= -------------------, 6 SQRT(79) %1 - 34

SQRT(79) %1 + 1

C = ---------------], [A 10

5 SQRT(79) %1 - 5 B

= -----------------, SQRT(79) %1 - 11

5 SQRT(79) %1 + 5

B =-----------------, SQRT(79) %1 + 11

25 SQRT(79) %1 - 25

= -------------------, 6 SQRT(79) %1 + 34 SQRT(79) %1 - 1

C

= - ---------------]] 10

Note that the display consists of a "list", i.e., some expression contained between two brackets [",], which itself contains two lists.

Each of the

latter contain a distinct solution to the simultaneous equations.

INTRODUCTION TO MACSYMA

Trig identities are easy to manipulate in MACSYMA.

221

The function TRIGEXPAND

uses the sum-of-angles formulas to make the argument inside each trig function as simple as possible:

(CIO) SIN(U+V)*COS(U)A3; 3

(DIO)

COS (D) SIN(V + D)

(Cll) TRIGEXPAND(%); 3

COS (D) (COS(U) SIN(V) + SIN(D) COS(V»

(DII)

The function

TRIGREDU~,

on the other hand. converts an expression into a form

which is a sum of terms, each of which contains only a single SIN or COS:

(CI2) TRIGREDDCE(DlO); SIN(V + 4 D) + SIN(V - 2 U) (Dl2) 8

3 SIN(V + 2 U) + 3 SIN (V) + -------------------------

8

The functions REALPART and lMAGPART will return the real and imaginary parts of a complex expression:

(CI3) W:3+K*%I; (Dl3) (C14) W 2*"-'E A

%1 K + 3 A

W; 2

(DI4)

(%1 K + 3)

%1 K + 3

%E

(CI5) REALPART(%); 3

(DI5)

%E

2

3

(9 - K ) COS(K) - 6 %E K SIN(K)

222

INTRODUcrION TO MACSYMA

Calculus

MACSYMA can compute derivatives and integrals. expand in Taylor series. take limits, and obtain exact solutions to ordinary differential equations. begin by defining the symbol F to be the following function of X:

(el) F:XA 3*%E A (K*X)*SIN(W*X);

3 KX X %E SINeW X)

(D!)

We compute the derivative of F with respect to X:

(C2) DIFF(F,X);

3 KX (D2) K X %E SINeW X)

+ 3

2 KX X %E SINeW X) 3 + WX

K X

%E

CDS(W X)

Now we find the indefinite integral of F with respect to X:

(C3) INTEGRATE(F,X);

6345273 (D3) «(K W + 3 K W + 3 K W + K ) X 6

24

42 62 W - 3 K ) X

+ (3 W + 3 K W - 3 K

4

32

5

4

22

+ (- 18 K W - 12 K W + 6 K ) X - 6 W + 36 K W 4 - 6 K )

KX %E SINeW X) +

«-

7

W -

2 3 K

5

W -

4 3 K

3

W

63 5 33 52 - K W) X + (6 K W + 12 K W + 6 K W) X 5 23 4 3 3 + (6 W - 12 K W - 18 K W) X - 24 K W + 24 K W)

8

26

44

CDS(W X»)/(W + 4 K W + 6 K W

628 + 4 K W + K )

A slight change in syntax gives definite integrals:

KX %E

We

INTRODUCTION TO MACSYMA (C4)

223

INTEGRATE(I/X~2,X,I,INF);

(M) I (C5) INTEGRATE(I/X,X,O,INF);

INTEGRAL IS DIVERGENT Next we define the symbol G in terms of F (previously defined in CI) and the hypebolic sine function, and find its Taylor series expansion (up to, say, order 3 terms) about the point X=O:

3

KX

SINeW X)

X %E (00)

4

SINH (K X) (C7) TAYLOR(G,X,O,3); 2

2

W WX (W K + W) X (D7)/T/ -- + --- -

(3 WK

2

4

3

K

K

3

4 6K

3 +

3

W) X + ..

3 6 K

The limit of G as X goes to 0 is computed as follows:

(CB) LIMIT(G,X,O); W

(DS) 4. K

MACSYMA also permits derivatives to be represented in unevaluated form (note the quote):

(09) 'DIFF(Y,X); dY (D9) dX

The quote operator in (09) means "do not evaluate". have obtained 0:

(CIO) DIFF(Y,X); (DIO)

o

Without it, MACSYMA would

INTRODUCTION TO MACSYMA

224

Using the quote operator we can write differential equations:

(C11) 'DIFF(Y,X,2)+'DIFF(Y,X)+Y; 2

dY

d Y

--- +

(Dll )

2 dX

Y

+ dX

MACSYMA's ODE2 function can solve some first and second order ODE's:

(CI2) ODE2(D11,Y,X): - X/2 (D12) Y

= %E

SQRT(3) X

SQRT(3) X

(%K1 SIN(---------) + %K2 OOS(---------))

2

2

Matrix Calculations

MACSYMA can compute the determinant, inverse and eigenvalues and eigenvectors of matrices which have symbolic elements (i.e., elements which involve algebraic variables.)

We begin by entering a matrix M element by element:

(C1) M:ENTERMATRIX(3,3); Is the matrix 1. Diagonal Answer I, 2, 3 or 4 4: Row 1 Column 1: 0: Row 1 Column 2: 1; Row 1 Column 3: A: Row 2 Column 1: 1; Row 2 Column 2: 0: Row 2 Column 3: 1; Row 3 Column 1: 1; Row 3 Column 2: 1; Row 3 Column 3: 0; Matrix entered. [0

2. Symmetric

1

A ]

0

1 ]

1

0 ]

[ (D1)

[1

]

[ [

3. Antisymmetric

]

Next we find its transpose, determinant and inverse:

4. General

INTRODUCTION TO MACSYMA

225

(C2) TRANSPOSE(M);

[ 0 1 [ [ 1 0 [ [ A

(D2)

] ] ] ] 0 ]

(C3) DETERMINANT(M); (D3)

A +

(C4) INVERT(M) .DETOUT; AI] ]

[ - 1 [ [ [

- A

[

A ] ] - 1 ]

1

(D4)

A+ 1

In (C4). the modifier DETOUT keeps the determinant outside the inverse.

As a

check. we multiply M by its inverse (note the use of the period to represent matrix multiplication):

(CS) RATSIMP(M.D4); [1 [ [0

(D5)

0

1

0] ] 0]

0

1]

[ [0

]

In order to find the eigenvalues and eigenvectors of M. we use the function EIGENVECTORS:

(el) EIGENVECTORS(M);

SQRT(4 A + 5) - 1 SQRT(4 A + 5) + 1 (D7) [[[- -----------------. ----------------2

- 1],

2

SQRT(4 A + 5) - 1

SQRT(4 A + 5) - 1 [1. 1. 1]]. [1.

2 A+ 2

-

-----------------]. 2 A + 2

SQRT(4 A + 5) + 1 SQRT(4 A + 5) + 1

[1. ----------------- -----------------], [1. - 1. 0]] 2A+2

2A+2

In D7. the first triple gives the eigenvalues of M and the next gives their respective multiplicities (here each is unrepeated).

The next three triples

INTRODUCTION TO MACSYMA

226

give the corresponding eigenvectors of M.

In order to extract from this

expression one of these eigenvectors. we may use the PART function:

(CS) PART(%.2); (DS)

SQRT(4 A + 5) - 1 [1. - ----------------2A+2

SQRT(4 A + 5) - 1 - -----------------] 2A+2

Programming in MACSYMA

So far we have used MACSYMA in the interactive mode. rather like a calculator. However. for computations which involve a repetitive sequence of commands like the ones we present in this book. it is better to execute a program.

Here we

present a short sample program to calculate the critical points of a function f of two variables x and y.

The program cues the user to enter the function f.

then it computes the partial derivatives f

x

and f ' and then it uses the y

x = f y = O. The program is written outside of MACSYMA with an editor. and then loaded into MACSYMA with

MACSYMA command SOLVE to obtain solutions to f

the BATaI command.

Here is the program listing:

CRITPTS(): =( PRINT("PROGRAM TO FIND CRITICAL POINTS").

F:READ("ENTER F(X.Y)"). PRINT("F =".F). EQS:[DIFF(F.X).DIFF(F,Y)]. UNK:[X.Y]. SOLVE(EQS.UNK»$

The program (which is actually a function with no argument) is called CRITPTS.

Each line is a valid MACSYMA command which could be executed from the

keyboard. and which is separated from the next command by a comma.

The partial

INTRODUCTION TO MACSYMA

227

derivatives are stored in a variable named EQS, and the unknowns are stored in UNK.

Here is a sample run:

This is EUNICE MACSYMA Beta Test Release 308.2. [copyright message ... ] (c1) BATCH(CRITPTS)$ [program is loaded in ... ] (c4) CRITPTSO; PROGRAM TO FIND CRITICAL POINTS

ENTER F(X,Y) %E (X 3+y 2)*(X+Y); A

A

A

2 F

= (y

y +

(d4) [[x

3 + x

x) %e

= 0.4588955685487001

= 0.4942017368275118 %i [x = 0.3589790871086935 -

y

%i + 0.3589790871086935.

- 0.1225787367783657].

0.4588955685487001 %i,

= - 0.4942017368275118 %i - 0.1225787367783657]. [x = 0.4187542327234816 %i - 0.6923124204420268, y = 0.455912070111699 - 0.869726269281412 %i], [x = - 0.4187542327234816 %i - 0.6923124204420268. y

Y

= 0.869726269281412

%i + 0.455912070111699]J

INTRODUCTION TO MACSYMA Partial List of MACSYMA functions

See the MACSYMA Reference Manual [26J for more information.

Online information

about any function is available through the command DESCRIBE(function name).

ALLROOTS(A) finds all the (generally complex) roots of the polynomial equation A, and lists them in NUMERical format (i.e. to 9 significant figures).

APPEND(A,B) appends the list B to the list A. resulting in a single list.

BATCH(A) loads and runs a BATCH program with filename A.

COEFF(A,B,C) gives the coefficient of B raised to the power C in expression A. C may be omitted if it is unity.

CONCAT(A,B) creates the symbol AB.

CONS(A,B) adds A to the list B as its first element.

DEMOIVRE(A) transforms all complex exponentials in A to their trigonometric equivalents.

DENOM(A) gives the denominator of A.

DEPENDS(A,B) declares A to be a function of B.

This is useful for writing

unevaluated derivatives, as in specifying differential equations.

DESOLVE(A.B) attempts to solve a linear system A of o.d.e. 's for unknowns B using Laplace transforms.

See the Manual for details.

INTRODUCTION TO MACSYMA

229

DETERMINANT(A) returns the determinant of the square matrix A.

DIFF(A.Bl.CI.B2,C2, ... ,Bn,Cn) gives the mixed partial derivative of A with respect to each Bi. Ci times. DIFF(A,B).

For brevity, DIFF(A,B,l) may be represented by

·DIFF( ... ) represents the unevaluated derivative. useful in

specifying a differential equation.

EIGENVALUES(A) returns two lists. the first being the eigenvalues of the square matrix A, and the second being their respective multiplicities.

EIGENVECTORS(A) does everything that EIGENVALUES does, and adds a list of the eigenvectors of A.

ENTERMATRIX(A.B) cues the user to enter an AxB matrix. element by element.

EV(A,BI,B2 .... ,Bn) evaluates A subject to the conditions Bi.

In particular the

Bi may be equations. lists of equations (such as that returned by SOLVE), or assignments. in which cases EV "plugs" the Bi into A.

The Bi may also be words

such as NUMER (in which case the result is returned in numerical format), DETOur (in which case any matrix inverses in A are performed with the determinant factored out). or DIFF (in which case all differentiations in A are evaluated, i.e. 'DIFF in A is replaced by DIFF).

For brevity in a manual

command (i.e., not inside a user defined function), the EV may be dropped, shortening the syntax to A,BI.B2, ... ,Bn.

EXPAND(A) algebraically expands A. over addition.

In particular multiplication is distributed

INTRODUCTION TO MACSYMA

230

EXPONENTIALIZE(A) transforms all trigonometric functions in A to their complex exponential equivalents.

FACTOR(A) factors A.

FREEOF(A,B) is true if the variable A is not part of the expression B.

GRIND(A) displays a variable or function A in a compact format.

When used with

WRITEFILE and an editor outside of MACSYMA, it offers a scheme for producing BATCH files which include MACSYMA generated expressions.

IDENT(A) returns an AxA identity matrix.

lMAGPART(A) returns the imaginary part of A.

INTEGRATE(A,B) attempts to find the indefinite integral of A with respect to B.

INTEGRATE(A.B.C.D) attempts to find the definite integral of A with respect to B taken from B=C to B=D.

The limits of integration C and D may be taken as INF

(positive infinity) or MINF (negative infinity).

INVERT(A) computes the inverse of the square matrix A.

KILL(A) removes the variable A together with all its assignments and properties from the current MACSYMA envirohment.

LIMIT(A,B,C) gives the limit of expression A as variable B approaches the value C.

The latter may be taken as INF or MINF as in INTEGRATE.

INTRODUCTION TO MACSYMA

231

LHS(A) gives the left hand side of the equation A.

LOADFILE(A) loads a disk file with filename A from the current default directory.

The disk file must be in the proper format (i.e. created by a SAVE

command).

MAKELIST(A,B,C,D) creates a list of A's (each of which presumably depends on B), concatenated from B=C to B=D.

MAP(A,B) maps the function A onto the subexpressions of B.

MATRIX(Al,A2 .... ,An) creates a matrix consisting of the rows Ai, where each row Ai is a list of m elements, [Bl,B2, .... BmJ.

NUM(A) gives the numerator of A.

ODE2(A,B,C) attempts to solve the first or second order ordinary differential equation A for B as a function of C.

PART(A,Bl,B2, ... ,Bn) first takes the Blth part of A. then the B2th part of that, and so on.

PLAYBACK(A) displays the last A (an integer) labels and their associated expressions.

If A is omitted, all lines are played-back.

See the Manual for

other options.

RATSIMP(A) simplifies A and returns a quotient of two polynomials.

REALPART(A) returns the real part of A.

232

INTRODUCTION TO MACSYMA

RHS(A) gives the right hand side of the equation A.

SAVE(A.BI.B2 ....• Bn) creates a disk file with filename A in the current default directory. of variables. functions. or arrays Bi.

The format of the file

permits it to be reloaded into MACSYMA using the LOADFILE command.

Everything

(including labels) may be SAVE'd by taking BI equal to ALL.

SOLVE(A.B) attempts to solve the algebraic equation A for the unknown B. iist of solution equations is returned. the form C

A

For brevity. if A is an equation of

= 0 it may be abbreviated simply by the expression C.

SOLVE([AI.A2 ....• An].[BI.B2, ... ,Bn]) attempts to solve the system of n poiynomial equations Ai for the n unknowns Bi.

A iist of solution equations is

returned.

STRINGOUT(A.BI.B2 ....• Bn) creates a disk file with filename A in the current default directory, of variables (e.g. labels) Bi. and is not reloadable into MACSYMA.

The file is in a text format

However the strungout expressions can

incorporated into a FORTRAN or BASIC program with a minimum of editing.

sUBsr{A.B,C) substitutes A for B in C.

SUM(A.B,C.D) sums expression A as B varies from C to D.

TAYLOR(A,B,C,D) expands A in a Taylor series in B about B=C, up to and including the term (B-C)AD.

MACSYMA also supports Taylor expansions in more

than one independent variable; see the Manual for details.

TRANSPOSE{A) gives the transpose of the matrix A.

INTRODUCTION TO MACSYMA

233 .

TRIGEXPAND(A) is a trig simplification function which uses the sum-of-angles formulas to simplify the arguments of individual SIN or COS's. E.g. TRIGEXPAND(SIN(X+Y»

gives COS(X) SIN(Y) + SIN(X) COS(Y).

TRIGREDUCE(A) is a trig simplification function which uses trig identities to convert products and powers of SIN and COS into a sum of terms. each of which contains only a single SIN or COS.

E.g .• TRIGREDUCE(SIN(X)A2) gives

(1 - COS(2X»/2.

TRIGSIMP(A) is a trig simplification function which replaces TAN. SEC. etc .. by their SIN and COS equivalents.

It also uses the identity

234

INTRODUCTION TO MACSYMA

Exercises

1.

Integrate (log(x»20, then check your result by differentiating it.

2.

Define z to be x + i Y (where i is the square root of -I, i.e. %1). and

then F to be sin(z2+ log(z».

a~

Then find the REALPART of F, call it R, and

a~

compute -:2 + -:2' which should be zero. ax ay

3.

Be sure to RATS IMP your answer.

Find the equation of the circle passing through the points: (-2,7), (-4,1),

(4,-5).

Hint: Look for the solution in the form

{x - a)2 + {y _ b)2 =

r

2

Use the given points to obtain 3 equations of the above form, and then SOLVE for the 3 unknowns a,b,r.

4.

Check the matrix identity (A B)-I= B- 1 A-I by computing both sides when

A

=

[1 2 3] 4 7

5 8

6 K

B

=

[-2o 1 0] 1 -2 1 1-2

Use ENTERMATRIX to define A and B, then INVERT both. multiplication is represented by a period.

5.

Remember that matrix

Be sure to RATSIMP your answer.

Find a particular solution to the following fourth order differential

equation: y' , " - y' " - y"

- y' - 2y = x5

Hint: Look for a solution in the form of a 5th degree polynomial in x with unknown coefficients. differential equation.

Check your answer by substituting it back into the

REFERENCES

1. Armbruster.D. O(2)-Symmetric Bifurcation Theory for Convection Rolls to appear in Physica D (1987) 2. Armbruster,D. Computer Algebra Programs for Dynamical Systems Theory to appear in Proc. Int. Symp. On Pattern Formation, Tuebingen 1986, Eds. Dangelmayr,G .. Guettinger,W .. Springer-Verlag. Heidelberg 3. Arnold.V.I. Mathematical Methods of Classical Mechanics Springer-Verlag, New York (1978) 4. Auchmuty.J.F.G. and Nicolis.G. Bifurcation Analysis of Non-linear Reaction-Diffusion Equations I Bull. Math. BioI. 37:323-365 (1975) 5. Bogdanov,R.I. Versa1 Deformations of a Singular Point on the Plane in the Case of Zero Eigenvalues Functional Analysis and Its Applications, 9:144-145 (1975) 6. Carr,J. Applications of Centre Manifold Theory Applied Mathematical Sciences. 35 Springer-Verlag, New York (1981) 7. Cary.J.R. Lie Transform Perturbation Theory for Hamiltonian Systems Physics Reports 79:129-159 (1981) 8. Cesari,L. Asymptotic Behavior and Stability Problems in Ordinary Differential Equations Ergebnisse der Mathematik und ihrer Grenzgebiete, 16 (Third edition) Springer-Verlag, New York (1971) 9. Chakraborty.T. Bifurcation Analysis of Two Weakly Coupled van der Pol Oscillators Ph.D. thesis, Cornell University (1987)

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10. Chakraborty,T. and Rand.R.H. The Transition from Phase Locking to Drift in a System of Two Weakly Coupled van der Pol Oscillators in Transactions of Fourth Army Conference on Applied Mathematics and Computing, ARO Report 87-1. pp.1003-1018 (1987) 11. Cushman.R. and Sanders.J.A. Nilpotent Normal Forms and Representation Theory of sl(2.ffi) Contemporary Math. 56:31-51 (1986) 12. Goldstein.H. Classical Mechanics Addison-Wesley. Reading (1980. 2nd edition) 13. Golubitsky,M. and Schaeffer,D.G. Singularities and Groups in Bifurcation Theory. Vol.I Applied Mathematical Sciences. 51 Springer-Verlag. New York (1985) 14. Guckenheimer.J. and Holmes.P. Nonlinear Oscillations. Dynamical Systems and Bifurcations of Vector Fields Applied Mathematical Sciences, 42 Springer-Verlag. New York (1983) 15. Guckenheimer,J. and Knobloch.E. Nonlinear Convection in a Rotating Layer: Amplitude Expansions and Normal Forms Geophys. Astrophys. Fluid Dyn. 23:247-272 (1983) 16. Hale,J.K. Ordinary Differential Equations Wiley, New York (1969) 17. Holmes,C. and Holmes.P.J. Second Order Averaging and Bifurcations to Subharmonics in Duffing's Equation J.Sound Vib. 78:161-174 (1981) 18. Holmes,C.A. and Rand,R.H. Coupled Oscillators as a Model for Parametric Excitation Mechanics Research Communications. 8:263-268 (1981) 19. Keith.W.L. and Rand,R.H. Dynamics of a System Exhibiting the Global Bifurcation of a Limit Cycle at Infini ty Int.J.Non-Linear Mechanics 20:325-338 (1985) 20. Kevorkian.J. and Cole.J.D. Perturbation Methods in Applied Mathematics Applied Mathematical Sciences. 34 Springer-Verlag. New York (1981) 21. Laure,P. and Demay.Y. Symbolic Computation and Equations on the Center Manifold preprint. Nice (1986) 22. Len,J. Nonlinear Parametric Excitation using Lie Transforms and Averaging Ph.D. thesis. Cornell University (1987)

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23. Len,J. and Rand,R.H. Nonlinear Parametric Excitation to appear in Transactions of Fifth Army Conference on Applied Mathematics and Computing (1987) 24. Lichtenberg,A.J. and Lieberman,M.A. Regular and Stochastic Motion Applied Mathematical Sciences, 38 Springer-Verlag, New York (1983) 25. Love,A.E.H. A Treatise on the Mathematical Theory of Elasicity Cambridge, University Press (1927, 4th edition) 26. MACSYMA Reference Manual, Version 11 (1986) prepared by the MACSYMA group of SYMBOLICS, Inc. 11 Cambridge Center, Cambridge, MA 02142 27. Minorsky,N. Nonlinear Oscillations Van Nostrand, Princeton (1962) 28. Month,L.A. On Approximate First Integrals of Hamiltonian Systems with an Application to Nonlinear Normal Modes in a Two Degree of Freedom Nonlinear Oscillator Ph.D. thesis, Cornell University (1979) 29. Month,L.A. and Rand,R.H. An Application of the Poincare Map to the Stability of Nonlinear Normal Modes J.Applied Mechanics, 47:645-651 (1980) 30. Month,L.A. and Rand,R.H. Bifurcation of 4:1 Subharmonics in the Nonlinear Mathieu Equation Mechanics Research Communications, 9:233-240 (1982) 31. Moon,F.C. and Rand,R.H. Parametric Stiffness Control of Flexible Structures in Proceedings of the Workshop on Identification and Control of Flexible Space Structures, Vol.II,pp.329-342 Jet Propulsion Laboratory Publication 85-29 California Institute of Technology, Pasedena (1985) 32. Nayfeh,A.H. Perturbation Methods Wiley, New York (1973) 33. Nicolis,G. and Prigogine,I. Thermodynamic Theory of Structure, Stability and Fluctuations Wiley, New York (1977) 34. Normand,C., Pomeau,Y. and Velarde,M.G. Convective Instability: A Physicist's Approach Rev. Mod. Phys. 49:581-624 (1977) 35. Rand,R.H. Computer Algebra in Applied Mathematics: An Introduction to MACSYMA Research Notes in Mathematics, 94 Pitman Publishing, Boston (1984)

238

REFERENCES

36. Rand.R.H. Derivation of the Hopf Bifurcation Formula Using Lindstedt's Perturbation Method and MACSYMA in Applications of Computer Algebra. R.Pavelle. ed .. 293-308 Kluwer Academic Publishers. Boston (1985) 37. Rand.R.H. and Holmes.P.J. Bifurcation of Periodic Motions in Two Weakly Coupled van der Pol Oscillators Int.J.Non-Linear Mechanics 15:387-399 (1980) 38. Rand,R.H. and Keith.W.L. Normal Form and Center Manifold Calculations on MACSYMA in Applications of Computer Algebra. R.Pavelle, ed .. 309-328 Kluwer Academic Publishers. Boston (1985) 39. Rand.R.H. and Keith,W.L. Determinacy of Degenerate Equilibria with Linear Part x·=Y. y'=Q Using MACSYMA Applied Mathematics and Computation 21:1-19 (1987) 40. Reiss.E.L. Column Buckling - An Elementary Example of Bifurcation in: Bifurcation Theory and Nonlinear Eigenvalue Problems Eds. Keller,J.B. and Antrnann.S. Benjamin. New York (1969) 41. Sanders,J.A. and Verhulst,F. Averaging Methods in Nonlinear Dynamical Systems Applied Mathematical Sciences 59 Springer-Verlag, New York (1985) 42. Sattinger,D.H. Topics in Stability and Bifurcation Theory Lecture Notes in Mathematics 309 Springer-Verlag, Berlin (1973) 43. Sattinger,D.H. Group Theoretic Methods in Bifurcation Theory Lecture Notes in Mathematics 762 Springer-Verlag. Berlin (1979) 44. Stoker,J.J. Nonlinear Vibrations Wiley Interscience, New York (1950) 45. Takens,F. Singularities of Vector Fields Publ.Math.Inst.Hautes Etudes Sci. 43:47-100 (1974) 46. SCRATCHPAD II Newsletter Vol.l, No.1, Sept.1985 Sutor ,R.S .. Ed. Computer Algebra Group IBM Watson Research Center Box 218 Yorktown Heights, NY 10598

INDEX

Programs are in CAPITAL letters, and their listings are underlined. action-angle variables 135 adjoint linear operator 161 amplitude equation 196,204 and Liapunov-Schmidt reduction 196 autonomous differential equation 2.24 AVERAGE 114-115.117-119,120-121.125-126,127-130 averaging 107 general form with respect to 107 mth order 127 theorem 109 Benard problem 196,206 bifurcation equation 158 stability of solutions 179 symmetry of 167.184,207 bifurcations fold 123 in algebraic equations 176 nilpotent 71,72 nondegenerate 156,161,176 of subharmonics 106,125,152,153 pitchfork 47,48.198 stationary 156 steady state, in systems of p.d.e.·s 181 boundary conditions Dirichlet 167,175,213,214 Neumann 167,175 zero-flux 167 Boussinesq approximation 197

INDEX

240 Boussinesq equations 197,206,209 13russe1ator 214 canonical transformations 133,153 center manifold 27,68,155,204

for systems with damping 35 vs. Liapunov-Schmidt reduction 176,177,179 and Lindstedt's method 39 CM 31,32-34,36,45

convection 196 damping 12,17,24,35,70,106

DECOMPOSE 65 determinacy 80-83 Taken's condition for 80 Duffing equation 24-25,99,120-123,131,154 averaging for 120 eigenfunction adjoint critical 161 critical 161 eigenvalue double zero 71 zero real part 27

EIGENVALUES 42 EIGENVECTORS 42 elastica 156,162 elliptic integrals 156,159 equations of motion 132 Euler buckling 156,162,168 feedback-control system 29 forcing T-periodic 107 Frechet derivative 160 Fredholm alternative 185 frequency-amplitude relation 4 generalized coordinates and momenta 133 generating function 133,134 Hamiltonian system 71,89,132 harmonic oscillator 133 two degree of freedom 149 Hartmann's theorem 53,72

INDEX homogeneous differential equation. solution of 167.175 Hopf bifurcation 17.21,37,70,85.89,109,156 stability formula for 20,67 HOPF2. HOPF3, '"

65-67

hyperbolic equilibrium point 52 Kami 1tonian 141 kernel 161 n-dimensional 205 LC 6,7-9 LC2 13.14-16,19-20.22-23,38-39 LIE 139-144,144-149,149-151 Lie transformation 132 vs. normal form transformation 132 Liapunov-Schmidt reduction 155 for degenerate problems 205 for higher order coefficients 205 for an integrodifferential equation 211 for o.d.e.'s 176 for p.d.e.'s 181,196 nonuniqueness of 179.212 via FORTRAN 206 and amplitude equations 196 vs. center manifold reduction 176,179 limit cycle 1 Lindstedt's method 1,37,49,50,85,86,89,132 for nonautonomous problems 24 and center manifold reduction 39 vs. normal form transformations 67, 70 Lorenz equations 41-49.177-180,196,202,203 MACSYMA,

algebra 219 ari thmetic 217 calculus 222 matrix manipulation 224 partial list of functions 228 special keys and symbols 216

INDEX

242

Mathieu equation 26,99,103-104,106,124,136-144,152,153 almost periodic solutions to 104 nonlinear 136 resonance instability for 104 transition curve 26 Navier Stokes equations 197,203 near-identity transformation 51,134,137 NF 55-60,63-67,74-79 nonautonomous differential equations 24,26,107,120,122,136 Lindstedt's method for 24 normal forms 50,107,204 Bogdanov's, for nilpotent bifurcations 73 Takens', for Hopf bifurcations 53 Takens', for nilpotent bifurcations 72,73 vs. Lie transformation 132 vs. Lindstedt's method 67,70 ODE2 175 oscillator systems of n

coup~ed

86-88,98-105,106,149-151

parametric excitation 26 particular solution 167,175 period doubling 125 Poisson bracket 134 range 161 Rayleigh-Benard problem 196-198,205-207 reaction diffusion equations 181,214 REDUCTION1 170-174,211,212,213 REDUCTION2 178-180,211 REDUCTION3 187-188,189-195,212 regular perturbation theory 4,26,124 resonances 142 resonant term 3,91,99,213 response function 154 SCRATCHPAD 181 secular terms 3,25,91,99 semilinear wave equation 182-186,212 slow flow equations 91,103-106 slow time scale 90 state variable 161

INDEX subcritica120.21.198 subharmonic solution 106.122.125.152.153 subharmonic resonance 112.142 supercritica1 20.21.198 symmetry of bifurcation equations 167.184.207 TRANSFORM 43-44.44-45.61.115-116.120.121 TWOVAR 93.94-98.100.102 two variable expansion 88.89-91 variation of parameters 111 van der Pol equation 1-7.24.49.71.85-88.89-93.112-116 at infinity 49.71 modified 119 n coupled 106 two coupled 86-88.105 van der Pol transformation 111-112 wedge product 153